Calc 2 Midterm Prep

Area of a washer

A = π(R² - r²), where R is the outer radius and r is the inner radius.

Area of a cylindrical shell

A = 2πrh, where r is the radius of the shell and h is the height.

Constant Force Formula

F = ma, where F is the force, m is the mass, and a is the acceleration.

Constant Work Formula

The work done by a constant force is given by W = Fd, where W is work, F is the force applied, and d is the displacement in the direction of the force.

Varying Force Formula

The work done by a varying force is calculated using the integral W = ∫ F(x) dx over the path of displacement. This accounts for changes in force applied during the movement.

Work Involving Objects made of parts (like liquids)

W = integral from a to b of weight density times cross sectional area at depth y times distance that slice at depth y is lifted.

Mean Value Theorem for Integrals

If f(x) is continuous on the interval [a, b], then there exists at least one c in [a, b] such that f(c) = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx.

Integration by Parts Formula

The Integration by Parts Formula states that ( \int u \, dv = uv - \int v \, du ), where ( u ) and ( dv ) are differentiable functions.

When to use Integration by parts?

Form: xⁿ e^x , xⁿ sinx, xⁿ cosx, xⁿ lnx, and inverse functions

For ∫sin^mcosⁿdx, do the following: (when n is odd)

Save one cos(x) and convert the rest to sin using cos^2x = 1-sin²x

For ∫sin^mcosⁿdx, do the following: (when m is odd)

Save one sin(x) and convert the rest to cos using sin²x = 1-cos²x

For ∫sin^mcosⁿdx, do the following: (if both m and n are even)

Use the identities sin² = 1-cos(2x) / 2

For ∫tan^msecⁿdx, do the following: (when n is even)

Save one sec²x and convert the rest to tan using sec²x = tan²x +1

For ∫tan^msecⁿdx, do the following: (when m is odd)

Save a secxtanx and convert the rest to sec using tan²x = sec²x -1ⁿⁿⁿ

For ∫tan^msecⁿdx, do the following: (when m is even and n is odd)

Convert everything to sec and integrate by parts with dv = sec²x

For ∫tanⁿxdx, do the following:

Convert one tan²x to sec²x -1 and split the problem into two integrals

For ∫secxdx, do the following:

ln|secx+tanx| + C

If the integrand involves √a²-b²x²

Use x = (a/b)sinθ

If the integrand involves √b²x² + a²

Use x = (a/b)tanθ

If the integrand involves √b²x² -a²

Use x = (a/b)secθ

Left-Hand Sum Integration

L_f = Δx Σ f(x_k - 1)

Right-Hand Sum Integration

R_f = Δx Σ f(x_k)

Trapezoid Rule Integration

_{Tf = (Δx/2) (f(a)+f(b)+ 2 Σ f(xk)}

Midpoint Rule Integration

M_f = Δx Σ f((x_k - x_k-1)/2)

Simpson’s Rule

S_f = (Δx/3) [f(x₀) +4f(x₁)+2f(x₂)+4(fx₂)+2f(x₃)+2f(x₄)+…+4f(x_n-1)+f(x_n)]

Trapezoid Error Rule

|E_T| <= (k(b-a)³/12n²) with |f”(x)| <= k on [a,b]

Midpoint Error Rule

|E_M| <= (k(b-a)³/24n²) with |f”(x)|<= k on [a,b]

Simpson’s Error Rule

|E_S| <= (k(b-a)⁵/ 180n⁴) with |f⁽⁴⁾(x)| <= k on [a,b]

integral from a to infinity of f(x) dx

lim as b goes to infinity of integral from a to b of f(x) dx

integral from c to a of f(x) dx

lim as b goes to a from the left of integral from c to b of f(x) dx