Integration Practice Examples

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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?

Isaac Newton Signup and view all the answers

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

[a, b] Signup and view all the answers

What term is used to describe functions that are differentiable?

Primitive functions Signup and view all the answers

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

x^2 + 1 Signup and view all the answers

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

(1/3)x^3 + x Signup and view all the answers

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

x^4 - 6x + C Signup and view all the answers

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

-cos(x) + sin(x) Signup and view all the answers

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

-cot(x) - cosec(x) + C Signup and view all the answers

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

tan(x) - sec(x) Signup and view all the answers

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

Parabolas Signup and view all the answers

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

$C$ Signup and view all the answers

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

Different parabolas with varying vertices Signup and view all the answers

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

It is shifted one unit along the y-axis in the positive direction Signup and view all the answers

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

$x = a$ intersects at P0, P1, P2, P-1, P-2 etc. Signup and view all the answers

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

Vertices move along the y-axis Signup and view all the answers

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

log x + 1 - log x + 2 Signup and view all the answers

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

When it is true for all (permissible) values of x. Signup and view all the answers

What is the proper rational function in Example 12?

x^2 - 5x + 6 Signup and view all the answers

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

To set up the integrand Signup and view all the answers

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

(x + 1)^2 Signup and view all the answers

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

An identity true for all permissible values of x Signup and view all the answers

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

b ∫a f(x) dx = lim h [f(a) + f(a + h) + ... + f(a + (n – 1)h] 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?

Dummy variable Signup and view all the answers

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

0 Signup and view all the answers

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

Integration Signup and view all the answers

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

'h→0' represents the limit of the interval width and 'n→∞' represents the number of intervals. 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?

Independence property Signup and view all the answers