10B Discussion Review Problems

A set of flashcards covering the key problem-solving strategies and concepts from the 10B Discussion Review Problems lecture notes.

What is the area of the region bounded by x = 2 − y²/4 and x = e^y − 2 between y = −2 and y = 2?

Find the area using integration between the specified bounds.

What is the volume of the solid obtained from rotating the curve y = x²−x−2 about the x-axis?

Calculate the volume using the method of disks or washers for solids of revolution.

How do you compute the integral ∫ x³/(x² + 1) dx?

This requires a form of polynomial long division and/or substitution.

What is the integral of tan x?

∫ tan x dx = -ln |cos x| + C.

How do you compute ∫ x arctan x dx?

Use integration by parts to solve this integral.

What result do you get from the integral ∫ 10x (x − 2)(x² + 1) dx?

Use polynomial expansion and then integrate term by term.

What is the integral of x⁵ (x⁶ + 5)^{3/2} dx?

Apply substitution to simplify the integral before integrating.

Do the improper integral ∫₄⁻² 1/x dx converge or diverge?

Diverges due to the behavior at x=0.

What is the convergence of the integral ∫₁^∞ ln(x)/x dx?

Converges, and its value can be computed using the limit as the upper bound approaches infinity.

How do you solve the initial value problem x²y'+2xy=6e^(3x), y(1) = −5 + 2e^(3)?

Rewrite the left side as the derivative using the product rule and solve.

What is the solution to the initial value problem y' = e^(x+y), y(0) = 0?

Use separation of variables technique to solve.