BSc 1st Semester Maths - Integral Calculus Past Papers PDF
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2017
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Summary
This document contains past papers from an undergraduate mathematics course covering integral calculus topics such as definite integrals, Riemann integrals, and improper integrals. The important questions are from the (2017-21) series but without a specific exam board or school.
Full Transcript
# B.Sc. Ist Semester Maths ## II. Book - Integral Calculus. ### Important Questions Series [2017-21] **Ch-Definite Integrals** * S.T: $\int_{0}^{\pi/2} log sinx dx = - \frac{1}{2}log 2$ * S.T: $\int_{0}^{\pi/4} log(1+ tan\theta) d\theta = \frac{\pi}{8}log 2$. * S.T: $\int_{0}^{\pi} \frac{tanx d...
# B.Sc. Ist Semester Maths ## II. Book - Integral Calculus. ### Important Questions Series [2017-21] **Ch-Definite Integrals** * S.T: $\int_{0}^{\pi/2} log sinx dx = - \frac{1}{2}log 2$ * S.T: $\int_{0}^{\pi/4} log(1+ tan\theta) d\theta = \frac{\pi}{8}log 2$. * S.T: $\int_{0}^{\pi} \frac{tanx dx }{sec^2 tanx} = \pi(\frac{\pi}{8} - 1)$ * S.T: $\int_{0}^{\pi/2} \frac {sin^2x dx}{sin^2x+cos^2x} = \frac{\pi}{4}$ ### Ch- The Riemann Integral * Define Lower and Upper Riemann Integrals. * Define R-Integrability. * A necessary and sufficient condition for R-integrability of a bounded function f: [a,b] → R over [a,b] is that for every ∈>0, ∃ a partition P of [a, b] such that for P and all its refinements, $0 ≤ U(P,f) -l(P,f) ≤ ∈$. * Let f(x) = 2x on [0,1]. Calculate $\int_{0}^{1} xdx$ and $\int_{0}^{1} xdx$ by dissecting [0,1] into n equal parts and hence S.T: f∈R[0,1]. * If a function f is defined on [0, a], a>0 by f(x) = x, then show that f is Riemann integrable on [0, a] and $\int_{0}^{a} f(x)dx = \frac {a^2}{4}$. * Fundamental theorem of Integral Calculus:- Let f∈R[a,b] and let φ be a differentiable function on [a, b] such that φ'(n)=f(x) for all x∈[a, b]. Then, $\int_{a}^{b} f(x)dx = φ(b)- φ(a)$ ### Ch- Differentiation Under the Sign of Integration जितना कराया है, complete करना है। ### Ch - Improper Integrals * Test the convergence of the integral $\int_{2}^{\infty} \frac {sin \pi x dx}{1+x^2}$ * Define Abel's Test. * S.T:- The integral $\int_{0}^{\pi/2} 1^2 log sinx dx$ converges. * Test the convergence of the following integrals: (i) $\int_{0}^{\infty} \frac{cos x dx}{1+x^2}$ (ii) $\int_{0}^{\infty} \frac{sin^3 x dx}{x(π^2+x^2)}$ ### Ch- Beta and Gamma Functions * Define Beta and Gamma function. * Show that: Symmetry of Beta function i.e., B(m, n) = B(n, m) * Express the following integral in terms of Beta function: $\int_{0}^{1} x^n \frac{dx}{\sqrt{1-x^{2s}}}$ * Relation between Beta and Gamma functions. B(m,n)= $\frac{\Gamma m \Gamma n }{\Gamma (m+n)}$ where m>0, n>0. * S.T: $\frac{\Gamma (1+n)}{\Gamma (1-n)} = \frac{\pi n}{sin \pi n }$ where 0<n<1 * The value of $\Gamma\frac{1}{2}$ = $\sqrt π$ * P.T * Duplication formula:- $\Gamma m \Gamma m+1 = \sqrt{\frac{π}{2^{2m-1}}} \Gamma 2m$, where m>0. * $\int_{0}^{\infty} \frac{dx}{\sqrt{1+x^2}}$ = $\sqrt \pi$ * P.T- B(m,n) = B(m+1,n) + B(m, n+1) for m≥0, n>0; * P.T- $\frac {B(m+1,n)}{B(m,n)}= \frac{m}{m+n}$ * P.T- $\int_{0}^{\infty} e^{-ax} x^{n-1} dx = \frac{\Gamma n}{a^n}$ * P.T- $\int_{0}^{1} x^{m-1} (1-x)^{n-1} dx = B(m,n) = \frac{\Gamma m \Gamma n}{\Gamma (m+n)}$, * S.T: $\int_{0}^{1} \frac{dx}{(1-x^2)^{1/2}} = \frac{\pi}{2} K\sqrt n$ * S.T: $\int_{0}^{\pi/4} tan^2 \theta d\theta = \int_{0}^{1} \frac {dx}{1+x^2}$ - $\int_{0}^{1} \frac{x^2 dx}{1+x^2}$ = $\frac{\pi}{4}-1$. ### Ch-Double and Triple Integral * Evaluate the double integrals: * $\int_{a}^{b} \int_{a}^{b} (x^2+y^2) dxdy$. * $\int_{0}^{1} \int_{0}^{2} xy(1+x+y) dx dy$ * $\int_{0}^{1} \int_{0}^{ \sqrt{1+x^2}}$ $\frac {dx dy}{1+x^2+y^2}$ * S.T: $\int_{0}^{2} \int_{0}^{9/2} y dy dx = \int_{0}^{9} \int_{0}^{\sqrt y} x dx dy$ * Evaluate $\int \int (x^2+y^2) dxdy$ over the region in the positive quadrant for which x+y≤1. * Evaluate the following double integrals: * $\int_{0}^{\infty} \int_{0}^{x^2} e^-ln dxdy$ * $\int_{0}^{1} \int_{\sqrt{3n-1}}^{2} x dxdy$ * $\int_{0}^{1} \int_{0}^{\sqrt{1-2^2}} \frac{dx dy}{(1-x^2)(1-y^2)}$ * Evaluate $\int \int x²y² dxdy$ over the region x²+y² ≤1. * Evaluate $\int \int xy dxdy$ over the region in the positive quadrant for which x+y≤1. * Evaluate $\int \int \frac{r^{2} dr d\theta }{a^2+2r^2}$ over one loop of the lemniscate r²= a²cos2θ. * Evaluate $\int_{}^{} \int_{}^{} r^3 sinθ d\theta dr$ * Evaluate $\int_{}^{} \int_{}^{} r^4 a(1+cos θ) d\theta dr$ * Evaluate SSS (x+y+z) dndydz over the tetrahedron x=0, y=0, z=0 and x+y+z=1. * Evaluate $\int_{0}^{1}\int_{ -1}^{- y}\int_{0}^{1-1-y} dxdydz$ * Evaluate $\int_{0}^{1}\int_{0}^{y} \int_{0}^{\sqrt{2y^2-x^2}} dz dxdy$ * Change the order of integration in the double integral $\int_{0}^{\infty} \int_{0}^{x} e^{-y} dx dy$ and hence find its value. * Change the order of integration in the following integrals: * $\int_{0}^{3} \int_{y}^{3} u^2y (nty) dydn$. * $\int_{0}^{2a} \int_{√2ax-x^2}^{ √ax} f(nx,y) dndy$ ### Ch-Dirichlet's and Liouville's Integrals * Evaluate $\int \int x^{2m-1} y^{2n-1} dxdy$ for all positive values of x and y such that x²+y² ≤c². * Find the value of $\int\int\int log(x+y+z) dxdydz$, the integral extending over all positive values of x, y, z subject to the condition x+y+z ≤1. * P.T:- when x and y are positive and x+y ≤ b $\int\int y^l(x+y)^{s-1} y^{-1-l} dxdy = \frac{k^2}{s} [B(s+l)- B(s+l+1)]$ * S.T:- The integral $\int\int\int x^{l-1} y^{m-1} z^{n-1} dxdydz$ integrated over the region in the first octant below the surface $(\frac {x}{a})^p + (\frac{y}{b})^q + (\frac{z}{c})^r=1$ is $\frac{a^lb^mc^n}{pqr} . \Gamma(\frac {l}{p}) \Gamma(\frac{m}{q}) \Gamma(\frac{n}{r})$. * Evaluate $\int\int\int dx dy dz$ where $\frac {x^2}{a^2} + \frac{y^2}{b^2} $$+ \frac{z^2}{c^2} ≤1$. * The plane $\frac {x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ meets the coordinate axes in the points A, B, C. Use Dirichlet’s integral to evaluate the mass of the tetrahedron OABC, the density at any point (x, y, z) being kxyz. * Evaluate SSS e^(x+y+z) dndydz taken over the positive octant such that x+y+z≤1. ### Ch-Rectification * Find the perimeter of the loop of the curve 3xy² = x²(a-x). * Find the length of the astroid x²/³ + y²/³ = a²/³. * Find the length of the curve y = loge(x+1) / (ex+1) from n=1 to n=2. * Find the length of the arc of the parabola y² = 4ax cut off by the line y = 3x. * Find the perimeter of the curve x²/a² + y²/b² = a². * S.T’:- 8a is the length of an arc of the cycloid whose equations are n = a(t-sint), y = a(1-cost) * Find the whole length of the astroid n = a cos³t, y = a sin³t. * Find the perimeter of the cardioid r= a(1-cosθ) * Find the entire length of the cardioid r = a (1+ cos θ) * Find the intrinsic equation of the equiangular spiral p = a e^(θ/√2) * Find the intrinsic equation of the catenary y = a cosh(x/a). ### Ch-Volumes and Surfaces of Solids * Find the volume of the solid generated by the revolution of the cissoid y²(2a-x) = x³ about its asymptote. * Find the volume of a spherical cap of height h cut off from a sphere of radius a. * P.T:- the volume of the solid generated by the revolution of an ellipse round its minor axis is a mean proportional between those generated by the revolution of the ellipse and of the auxiliary circle about the major axis. * Find the volume formed by the revolution of the loop of the curve y² (a+x) = 2(a-x) about the axis of n. * The cardioid r = a(1+cos θ) revolves about the initial line. Find the volume of the solid thus generated. * S.T:- The volume of the solid formed by the revolution of the curve r = a+bcos θ (a>b) about the initial line is πr²a (a²+b²). * Find the volume of the solid generated by revolving one loop of the lemniscate r²= a² cos 2θ about the line θ = π / 4. * Find the surface generated by the revolution of an arc of the catenary y = a cosh(x/a) about the axis of n. * Find the surface of the solid generated by revolution of the astroid x²/³ + y²/³ = a²/³ with x = a cos³t and y = a sin³t about the n-axis. * State Pappus and Guldin theorem. * S.T:- The volume generated by the revolution of the ellipse x²/a² + y²/b² = 1 about the line x = 2a is 4π²a²b. ### Ch-Differentiation of Vectors * S.T- The necessary and sufficient condition for the vector a(t) to have constant magnitude is $\frac{d}{dt}$ a.a = 0. * S.T- The necessary and sufficient condition for the vector a(t) to have constant direction is a x $\frac {d}{dt}$a = 0. * If a = a cost i + a sin t j + a tan t k, find $|\frac{d}{dt}$a x $\frac{d^2}{dt^2}$a$| and $|\frac{d}{dt}$a, $\frac{d^2}{dt^2}$a, $\frac{d^3}{dt^3}$a$|. * If $\frac{d}{dt}$u = wxu, $\frac{d}{dt}$v = wxv, S.T:- $\frac{d}{dt}$ (uxv) = w x(uxv) * A particle moves along the curve x = 4 cost, y = 4 sint, z = 6t, find the velocity and acceleration at time t = 0 and t = π/2. Find also the magnitudes of the velocity and acceleration at any time t. * If $\int_{}^{} f\times d^2 t = 0$, S.T:- f = constant. * If r = a + bt, then find $\frac{d^2}{dt^2}$r = 0. * If r = (a+bt)² + (c + dt)² + (e + ft)², where a, b, c, d, e, f are constants, S.T:- $\frac{d^2}{dt^2}$r = 0. ### Ch- Gradient, Divergence and Curl * If r = |r| r where r = xi + yj + zk, P.T:- * (i) ∇(|r|) = $\frac{r}{|r|}$ * (ii) ∇(r) x r = 0 * (iii) ∇($\frac{1}{|r|}$) = - $\frac{r}{|r|^3}$ * (iv) ∇($\frac{1}{r^2}$) = - $\frac {2r}{r^4}$ * (v) ∇ log |r| = $\frac{r}{|r|^2}$ * (vi) ∇.r = n+m+n = 3 * Find grad f, where f is given by f = x³-y³+x²z², at the point (1, -1, 2) * P.T:- ∇T³ = -3T²∇T * S.T:- grad(r.∇a) = ∇a. * Find a unit normal vector to the level surface xy+2nz = 4 at the point (2, -3, 3) * Find the directional derivative of f(x, y, z) = xyz + ynz² at the point (1, 2, -1) in the direction of the vector 2i-j+√2k. * Find the maximum value of the directional derivative of φ = x²yz at the point (1, 4, 1). * P.T:- div A = 3. * P.T:- Curl A = 0. * If F = xyj - 2xz² i + 2 y²zk find * (i) div F * (ii) curl F * (iii) curl curl F * If a is a constant vector, find * (i) div (r x a) * (ii) curl (r x a) * If u = x²-y²+4z, S.T:- ∇u = 0 * S.T:- The vector F = 3y²z² i + 4x³z² j - 3xy²zk is solenoid but not irrotational. * P.T:- curl (φA) = (grad φ) x A+ φ curl A * P.T:- curl(AXB) = (B.∇)A - B div A - (A.∇)B + A div B * P.T:- curl curl A = 0. i.e., ∇.(∇ x A) = 0 * P.T:- div curl A = 0 * P.T:- div (r²) = (n+3) |r| * P.T:- div ($\frac{A}{|r|}$) = 0 * P.T:- ∇²f(r) = f''(r) + $\frac {2}{r}$ f'(r) * P.T:- ∇²f(r) = 0 or div (grad f) = 0. * P.T:- div grad sin r = n(n+1) r^(n-2) i.e., ∇²r^n = n(n+1) r^(n-2) ### Ch- Integration of Vectors Complete करना है [Examples + Exercise] ### Ch- Line Integrals * Evaluate $\int_{c} \overrightarrow{F}$.d$\overrightarrow{r}$, where $\overrightarrow{F}$ = xi + y²j and curve c is the arc of the parabola y = x² in the x-y plane from (0,0) to (1,1) * Evaluate $\int_{c} \overrightarrow{F}$.d$\overrightarrow{l}$ along the curve x²+y²=1, z=1 in the positive direction from (0,1,1) to (1,0,1), where $\overrightarrow{F}$ = (2x+yz)i + 2y²j + (2xy+z²)k * Evaluate $\int_{c} (xdy-ydn)$ around the circle x²+y²=1. * Evaluate $\int_{c} \overrightarrow{ F}$.d$\overrightarrow{r}$, where $\overrightarrow{F}$ = (x²+y²) i -2xy j and curve c is the rectangle in the x-y plane bounded by y = 0, x = a, y = b, x = 0. * Evaluate $\int_{c} \overrightarrow{F}$.d$\overrightarrow{r}$, where $\overrightarrow{F}$ = i cos y - j sin y and c is the curve y = 5x-x² in the x-y plane from (1, 0) to (0,1) * If F = (2x²-3z)i - 4y²j - ynt, then Evaluate $\int\int\int_{v} \overrightarrow{F}$.d$\overrightarrow{V}$ where v is the closed region bounded by the planes x = 0, y = 0, z = 0 and x + 2y + z = 4. Also Evaluate $\int\int\int_{v} \overrightarrow{D}\times \overrightarrow{F}$ d$\overrightarrow{V}$. ### Ch-Green's Gauss's and Stokes' theorem. * Define Green's Theorem * Define Gauss's Theorem. - Only Statement, Proof Syll में नहीं है। * Define Stoke's Theorem. - Only Statement, Proof Syll में नहीं है। * Evaluate by Green's theorem (x²-cashy)dx + (y + sinx) dy, where c is the rectangle with vertices (0, 0), (π, 0), (π, 1), (0, 1). * Evaluate by Green's theorem (cos x sin y - xy)dx + sin x cos y dy, where c is the circle x² + y² = 1 * For any closed surface S, P.T.: $\int\int_{^{}} curl \overrightarrow{F}$ . d$\overrightarrow{s}$ = 0. * If $\overrightarrow{F}$ = ax i + by j + cz k, a, b, c are constants, S.T: $\int\int_{^{}} \overrightarrow{F}$.d$\overrightarrow{s}$ == 4π (a+b+c), where s is the surface of a unit sphere. * Evaluate $\int\int\int_{}^{} x^2dydz + y^2dzdx + z^2(xy - x - y ) dxdy$ where s is the surface of the cube 0≤x≤1, 0≤y≤1, 0≤z≤1. * If $\overrightarrow{F}$ = axi + by j + cz k, where a, b, c are constants, S.T:- $\int\int\int_{^{}} (\overrightarrow{D}.\overrightarrow{F})$ ds = 4π (a+b+c), S being the surface of the sphere (x-1)² + (y-2)² + (z-3)² = 1. * Verify Stoke's Theorem for $\overrightarrow{F}$ = y² i + z² j + x² k where s is the upper half surface of the sphere x²+y²+z²=1 and c is the boundary. * Evaluate by Stoke's Theorem $\int_{c} (yzdx + xzdy + xy dz)$ where c is the curve x²+y²=1, z = y².
