Integration

Question 1

What is an indefinite integral?

Answer 1

Antiderivative 
If f(x) = g’(x), then ∫ f(x) dx = g(x)
The integral of f(x) with respect to x is equal to g(x).

Question 2

Constant Multiple

Answer 2

∫ k • f(x) dx = k ∫ f(x) dx

Ex: ∫ 2 cos x dx = 2 ∫ cos x dx

Question 3

Sum/Difference

Answer 3

∫ [ f(x) ± g(x) ] dx = ∫ f(x) dx ± ∫ g(x) dx

Ex: ∫ [ 7 + e^x ] dx = ∫ 7 dx + ∫ e^x dx

Question 4

Mean Value Theorem

Answer 4

If f is a function that is continuous on [a, b] and differentiable on (a, b), then there is a number c on the ineterval [a, b] such that f’(c) = [ f(b) - f(a) ] / (b - a)
f(b) - f(a) = f’(c)(b - a)

Question 5

Rolle’s Theorem

Answer 5

If f is a function that is continuous on [a, b] and differentiable on (a, b), and f(a) = f(b), then there is a number c on the ineterval [a, b] such that f’(c) = 0

Question 6

The Fundamental Theorem of Calculus

Answer 6

If f is a function that is continuous on [a, b], and f(x) = F’(x), then ∫𝑎𝑏 f(x) dx = F(b) - F(a)

Question 7

The Definite Integral

Answer 7

If f is both positive and negative, then the definite integral represents the net or signed area, i.e. the area above the x-axis and below the graph of f minus the area below the x-axis and above the graph of f.

Question 8

Reverse Interval

Answer 8

∫ a to b f(x) dx = -∫ b to a f(x) dx

Question 9

Zero-length Interval

Answer 9

∫ a to a f(x) dx = 0

Question 10

Adding Intervals

Answer 10

∫ a to b f(x) dx + ∫ b to c f(x) dx = ∫ a to c f(x) dx

Question 11

Derivative vs Integral

Answer 11

The derivative gives the slope of the function at a point. The integral gives the area under the curve between two points.

Question 12

Volume of Revolution: Disk Method

Answer 12

If f(x) is continuous and f(x) ≥ 0 on [a, b], then the solid obtained by rotating the region under the graph about the x-axis has volume [with R = f(x)]
V = π ∫ a to b R^2 dx = π ∫ a to b f(x) ^2 dx

Question 13

The Washer Method

Answer 13

The disk method can be extended to cover solids of revolution with holes. The washer is formed by revolving a rectangle about an axis. If r and R are the inner and outer radii of the washer and dx is the width of the washer, then the volume is: ∫ a to b ([R(x)]^2 - [r(x)]^2) dx

Question 14

Volume of a Solid

Answer 14

The definition of a solid of unknown integrable cross section are A(x) from x = a to x = b is the integral of A from a to b, V = ∫ a to b A(x) dx

Question 15

Integrate with respect to x when:

Answer 15

the cross section is perpendicular to the x-axis

V = ∫ a to b A(x) dx

Question 16

Integrate with respect to y when:

Answer 16

the cross section is perpendicular to the y-axis

V = ∫ a to b A(y) dy

Question 17

The Fundamental Theorem of Calculus part 2

Answer 17

If g(x) = ∫ a to x f(t) dt, where a is a constant, then g’(x) = f(x)