Integration Practice Examples

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Questions and Answers

What was the original motivation for the concept of the derivative in Differential Calculus?

  • Defining and calculating tangent lines to graphs of functions (correct)
  • Studying the slope of tangent lines to graphs of functions
  • Determining the anti-derivatives of functions
  • Calculating the area of regions bounded by functions

What type of functions are considered as anti-derivatives in Integral Calculus?

  • Functions that could potentially have a given function as a derivative (correct)
  • Functions with constant derivatives
  • Functions that can be integrated easily
  • Functions with undefined derivatives

What is the main problem that Integral Calculus aims to solve?

  • Calculating the rate of change of functions
  • Defining and calculating the area enclosed by functions (correct)
  • Defining and calculating tangent lines to graphs of functions
  • Finding the maximum and minimum values of functions

Who is credited with the original motivation for the concept of the derivative?

<p>Isaac Newton (C)</p> Signup and view all the answers

In what interval must a function be differentiable for the derivative to exist at each point?

<p>[a, b] (D)</p> Signup and view all the answers

What term is used to describe functions that are differentiable?

<p>Primitive functions (D)</p> Signup and view all the answers

What is the result of the integral ∫ x^3 - 1 dx / x^2?

<p>x^2 + 1 (D)</p> Signup and view all the answers

For the integral 2 ∫ (x^3 + 1) dx, what is the correct result?

<p>(1/3)x^3 + x (A)</p> Signup and view all the answers

What is the anti-derivative of f(x) = 4x^3 - 6?

<p>x^4 - 6x + C (B)</p> Signup and view all the answers

What is the result of ∫ (sin x + cos x) dx?

<p>-cos(x) + sin(x) (A)</p> Signup and view all the answers

For the integral ∫ cosec x (cosec x + cot x) dx, what is the correct anti-derivative?

<p>-cot(x) - cosec(x) + C (D)</p> Signup and view all the answers

What is the result of ∫ (1 - sin(x)) / sin(x) dx?

<p>tan(x) - sec(x) (A)</p> Signup and view all the answers

What geometric shape do the integrals of the function $f(x) = 2x$ represent?

<p>Parabolas (B)</p> Signup and view all the answers

Which parameter results in different members of the family of integrals?

<p>$C$ (B)</p> Signup and view all the answers

What does assigning different values to $C$ in $y = x^2 + C$ achieve?

<p>Different parabolas with varying vertices (C)</p> Signup and view all the answers

For $C = 1$, how is the parabola $y = x^2 + 1$ related to $y = x^2$?

<p>It is shifted one unit along the y-axis in the positive direction (A)</p> Signup and view all the answers

If the line $x = a$ intersects various parabolas, when will the value of $dy$ at these points equal $2a$?

<p>$x = a$ intersects at P0, P1, P2, P-1, P-2 etc. (A)</p> Signup and view all the answers

What happens to the vertex location of parabolas in the family when different values of $C$ are assigned?

<p>Vertices move along the y-axis (B)</p> Signup and view all the answers

What is the integrand of ∫ (x + 1)(x + 2) dx?

<p>log x + 1 - log x + 2 (A)</p> Signup and view all the answers

In the context of the text, when is a statement considered an identity?

<p>When it is true for all (permissible) values of x. (D)</p> Signup and view all the answers

What is the proper rational function in Example 12?

<p>x^2 - 5x + 6 (D)</p> Signup and view all the answers

In Example 12, how is the equation A + B = 5 used?

<p>To set up the integrand (B)</p> Signup and view all the answers

What is the integrand of ∫ (x + 1)^2 (x + 3) dx?

<p>(x + 1)^2 (B)</p> Signup and view all the answers

What does the symbol '≡' signify according to the text?

<p>An identity true for all permissible values of x (C)</p> Signup and view all the answers

What is the expression known as the definition of definite integral?

<p>b ∫a f(x) dx = lim h [f(a) + f(a + h) + ... + f(a + (n – 1)h] (C)</p> Signup and view all the answers

What is the term used for the variable of integration that is not relevant to the value of the definite integral?

<p>Dummy variable (C)</p> Signup and view all the answers

In Example 25, what is the value of 'a'?

<p>0 (D)</p> Signup and view all the answers

What does the symbol '∫' represent in mathematical notation?

<p>Integration (B)</p> Signup and view all the answers

What do the limits h→0 and n→∞ represent in the context of definite integrals?

<p>'h→0' represents the limit of the interval width and 'n→∞' represents the number of intervals. (D)</p> Signup and view all the answers

What property ensures that the value of a definite integral depends on the function and the interval but not on the variable of integration?

<p>Independence property (D)</p> Signup and view all the answers

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