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Subject Code/Title: MAZION Breor Haebra Unit: T ## Unit - II - Integral Calculus ### Multiple Integration: Double and Triple Integrals - Change of order of Integration in double Integrals - Choice of Variables [Cartesian to Polar] - Volume of Solids - Gradient - Curl - Divergence - Theorems of Green...

Subject Code/Title: MAZION Breor Haebra Unit: T ## Unit - II - Integral Calculus ### Multiple Integration: Double and Triple Integrals - Change of order of Integration in double Integrals - Choice of Variables [Cartesian to Polar] - Volume of Solids - Gradient - Curl - Divergence - Theorems of Green’s in a plane - Gauss and Stokes theorems [Excluding Proof]. ### Scalable Integration [Cartesian Co-ordinates] #### 1. Evaluate ∫∫ x(x + y)dydx * Go: ∫∫ x(x + y)dydx = I * G1: ∫∫ x(x + y)dydx = I * = [x2 + xy2]dy = [x2 + xy2]01 * = [ x2 + x/1]dy = x2 + x/1 * = ( x2/2 + x/2 ) 0 - (0 + 0) = x2/2 + x/2 * = ( x2/2 + x/2) dy = x2/2 + x/2 * = ( x2/2 + x/2 ) dy = x2/2 + x/2 #### 2. Evaluate ∫∫ (x2 + y2)dxdy * Go: ∫∫ (x2 + y2)dxdy = I * G1: ∫∫ (x2 + y2)dxdy = I * = (x2 + y2)dx = [x3/3 + xy2]01 * = (1/3 + y2)dy = (1/3 + y2)dy * = (1/9 + y3/3)01 = 1/9 + 1/3 = 4/9 * = (x2/2 + y2)dy = [x2/2 + y2] * = (1/2 + y2)dy = (1/2 + y2) dy #### 3. Evaluate ∫∫ y2dxdy * Go: ∫∫ y2dxdy = I * G1: ∫∫ y2dxdy = I * = y2dx = [xy2]b1 * = [yb2] y=0b = [yb2] log b/log 1 * = [yb2] = log b Subject Code/Title: Unit: TYPE: I: Limits are Variables #### 1. Evaluate: ∫∫ (x2 + y2)dydx * Go: ∫∫ (x2 + y2)dydx = I [Correct Form] * =∫ [xy + y3/3] 02 * =∫ (2x + 8/3) dx * = [x3/3 + 4x/3] 05 = 125/3 + 20/3 = 145/3 #### 2. Evaluate ∫∫ ydydx * Go: ∫∫ ydydx = I [Correct Form] * =∫[y2/2] 1-xdx * =[1/2 - (1-x)2/2] dx * =∫[1 - (1-x)2/2] dx * = -1/2 [(1-x)3/3]01 * = -1/6 (0 - 1) = 1/6 * = 1/2 [(1-x)2/2]01 = 1/4 - 0 #### 3. Evaluate: ∫∫ y2dxdy * Go: ∫∫ y2dxdy = I [Correct Form] * =∫[xy2/2]01 * =∫[y2/2] log 1 - log 2 * = [y2/2] log 1 - log2 = [1/2] log b * = [y2/2] log 1 - log2 = [1/2] log 1 = 0 #### Evaluates: ∫∫ x/y *dxdy * Go: ∫∫ x/y * dxdy = I ⇔ * =∫[x2/2y] 01 * =∫[1/2y] log 1/b - log 1/a = [1/2y] log 1/b - log 1/a * =∫[1/2y] log 1/b - log 1/a = [1/2y] log 1/b - log 1/a = [1/2y] log 1/b * =∫[1/2y] log 1/b - log 1/a = [1/2y] log 1/b - log 1/a = [1/2y] log 1/b * = [1/2y] log 1/b - log 1/a = [1/2y] log (a/b) #### Evaluates: ∫∫ xy2dxdy * Go: ∫∫ xy2dxdy = I ⇔ * =∫[x2y2/2]01 * =∫[y2/2] log 1 - log2 * = [y2/2] log 1 - log2 = [1/2] log 1 = 0 * = [y2/2] log 1 - log 2 = [1/2] log 1 = 0 Subject Code/Title: Unit: * w.r.t 'y' keeping 'x' as a constant then consider the horizontal ship and the new limits accordingly. * w.r.t 'x' keeping 'y' as a constant then consider the vertical ship and the new limits accordingly. * After finding the new limits evaluate the inner integral first and then the outer integral. #### 1. Change the order the integration and hence evaluate it, for ∫∫e-ydydx * Go: ∫∫e-ydydx = I [Correct Form] * The region is bounded by x = 0; x = ∞; y = x; y = ∞ * x values from x:0 to x=∞ * y varies from y=x to y=∞ * After changing the order of integration, we've to integrate w.r.t 'x' first is then w.r.t 'y'. * ∫∫e-ydydx = ∫∫ e-y dxdy * = ∫ e-y [x]0∞ dy = ∫ e-y [∞ - 0] dy = ∫ e-y [∞] dy * = ∫ e-y [∞] dy =∫ e-y [∞] dy = ∫ e-y [∞] dy = ∫ e-y [∞] dy = ∫ e-y [∞]dy * = ∫ e-y [∞] dy = ∫ e-y [∞] dy = ∫ e-y [ ∞] dy = ∫ e-y [∞] dy = ∫ e-y [ ∞] dy = ∫ e-y [ ∞ ] dy = ∫ e-y [ ∞] dy = ∫ e-y [ ∞ ] * = ∫ e-y [ ∞] dy = ∫ e-y [ ∞ ] dy = ∫ e-y [ ∞] dy = ∫ e-y [ ∞] dy = ∫ e-y [ ∞] dy = ∫ e-y [ ∞] dy = ∫ e-y [ ∞ ] dy = ∫ e-y [∞] dy = ∫ e-y [∞] dy = ∫ e-y Subject Code/Title: Unit: #### 2. Change the order of Integration & then evaluate ∫∫ x2dxdy * Go: ∫∫ x2dxdy = I [Correct Form] * x Values from x = y to x = a * y Values from y = 0 to y = a * After changing the order of integration, we integrate w.r.t y first, then w.r.t x. * ∫∫ x2dxdy = ∫∫ x2dydx * = ∫ x2 [y]0a dx * = ∫ x2 [a] dx = a ∫ x2 dx * = a ∫ x2 dx = a ∫ x2 dx =a [ x3/3 ]0a * = a [x3/3]0a = a [ a3/3 - 0] = [a4/3] * = ∫[x3/3]0a = ∫[x3/3]0a = ∫[x3/3]0a = ∫[x3/3]0a = ∫[x3/3]0a = ∫[x3/3]0a = [ x3/3]0a #### 3. Change the order of Integration & evaluate ∫∫ xy2dxdy. * Go: ∫∫ xy2dxdy = I [Correct Form] * x Varies from ० to 4a * y varies from y = x2/4a to y = 2√ax * After changing the order of integration, we integrate w.r.t 'x' first and then w.r.t 'y'. * Hence ∫∫xy2dxdy * = ∫∫ xy2dydx = ∫√2ax x2/4a xy2dydx = ∫√2ax x2/4a xy2dydx * = ∫ [xy3/3] x2/4a 2√ax dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy =∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 * a√ay - y3/3] * x2/4a * dy * = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy * = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy * = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy =∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 * a√ay - y3/3] * x2/4a * dy * = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy * = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy * = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy =∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 * a√ay - y3/3] * x2/4a * dy * = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy * = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy * = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy =∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 * a√ay - y3/3] * x2/4a * dy * = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy * = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy * = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy * = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy =∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 * a√ay - y3/3] * x2/4a * dy * = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy * = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy * = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy =∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 *a√ay - y3/3] x2/4a dy = ∫ [8/3 * a√ay - y3/3] * x2/4a * dy * = ∫ [8/3 *a√ay - y3/3] * x2/4a * dy * = ∫ [y2/3] x2/4a 2√ax dy = y2/3 * [8/3 *a√ay - y3/3] x2/4a * = **y2/3 [8/3 *a√ay - y3/3] x2/4a** * = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy * = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy * = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy * = ∫y2/3 * [8/3 *a√ay - y3/3] * x2/4a * dy * = ∫y2/3 * [8/3 *a√ay - y3/3] * x2/4a * dy * = ∫y2/3 * [8/3 *a√ay - y3/3] * x2/4a * dy = ∫y2/3 * [8/3 *a√ay - y3/3] * x2/4a * dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy * = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy * = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy * = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy * = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a * dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a * dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a * dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a * dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a dy * = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a * dy = * = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a * dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a * dy * = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a * dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a * dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a * dy * = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a * dy = ∫y2/3 * [8/3 *a√ay - y3/3] x2/4a * dy = ∫y2/3 * [8/3 *a