INTEGRAL CALCULUS

What is an indefinite integral?

An integral without limits of integration

The process of finding the antiderivative of a function is known as:

Integration

What does the constant of integration C signify?

It accounts for the family of possible antiderivatives

Solve ∫(sin 2x)/(cos x) dx

−2 cos x + C

Evaluate ∫ sec² x dx

tan x + C

FTC Part 1 states:

The derivative of an antiderivative returns the original function

If f(x)=∫(x to x²) √t dt, find f'(x)

2x² − √x

Solve d/dx {∫(x² to x³) 1/(ln t) dt }

(x³ − x²)/ln x

FTC Part 2 states:

∫ₐᵇ f(x) dx = F(b) − F(a)

The definite integral ∫ₐᵇ f(x) dx is interpreted as:

The area under the curve between a and b

Evaluate ∫₃⁶ dx/x

ln 2

The method of substitution in integration is also known as:

u-substitution

Which technique is MOST appropriate for ∫(2x+3)e^(x²+3x) dx?

Algebraic substitution

After substitution, ∫2x(x²+1)⁴ dx becomes:

∫u⁴ du

After substitution, ∫x³(x²+1)² dx becomes:

(1/2) ∫u² du

Formula for integration by parts:

∫u dv = uv − ∫v du

When is integration by parts most appropriate?

When the integrand is a product of two functions

In ∫x² sin(3x) dx, the appropriate u is:

x²

Most appropriate technique for ∫(3x+2)/(x²−x−6) dx

Partial fraction decomposition

Most appropriate technique for ∫√(9−x²)/x² dx

Trigonometric substitution

Best substitution for ∫√(9−x²)/x² dx

x = 3 sin θ

An improper integral involves:

Both infinite limits and discontinuities

Why is ∫₀¹ 1/√x dx improper?

Discontinuity at x = 0

Why is ∫₀² 1/(1−x)² dx improper?

Vertical asymptote at x = 1

Evaluate ∫₂⁵ dx/√(x−2)

3√2

Evaluate ∫₋∞⁺∞ dx/(1+x²)

π

For ∫₁∞ 1/xᵖ dx, convergence occurs when:

p > 1

Evaluate ∫₁∞ 1/t¹·² dt

5

Evaluate ∫₀² 1/√|x−1| dx

Diverges

Area bounded by y = x³, x = −2 to 1

5.24

Area between y = sin x from 0 to 4π

8

Area under √x + √y = 1 in first quadrant

1/6

Area between y = x³, y = −x, y = 1

5/4

Area bounded by 4x − y² = 0 and y = 2x − 4

8

Formula for arc length of y = f(x)

∫√(1+(f'(x))²) dx

Arc length of y = x² + 2x from 0 to 1

3.17

Arc length of xy = 1 from (1,1) to (2, 1/2)

1.245

Which is NOT a solid of revolution method?

Euler method

Volume from rotating y = x³ and y = 1, 0≤x≤1 about x-axis

5π/6

Volume rotating region between y=x and y=x² about y=2

8π/15

Volume of region y²=12x, x=3 about x=3

181

Volume rotating y=2x²−x³ and y=0 about y-axis

16π/5

Volume rotating y = x − x² and y=0 about x=2

π/4

Average value of 3t³−t² on [−1,2]

11/4

Average amount of drug A(t)=30/(t+1)² over first 4 hours

6 mg

Average value of cos x on [−π/2, π/2]

2/π

Evaluate ∫₀³ ∫₁² x²y dy dx

27

Evaluate ∭ z dV over box 0≤x≤2, 0≤y≤3, 0≤z≤4

48

Evaluate ∫ (10y dA) over 0≤x≤1, 0≤y≤1−x

5/3

Find A = ∬(x+y) dA over 0≤x≤1, 0≤y≤2

3

Geometric meaning of a double integral:

Volume under a surface over a region