Calculus II

Use this for “Evaluate the integral of x sin(2x) from 0 to π/2.”

Use this for “Evaluate the integral of 1 ÷ ((x − 3)(x + 1)) from 4 to 6.”

Use this for “Evaluate the integral of √(9 − x²) from 0 to 2” (r = 3).

Use this for “Which method evaluates ∫ x·cos x dx?”

Use this for “Does the integral of 1 ÷ ∜(x³) from 0 to 1 converge? If so, to what?” (p = 3/4).

Use this for “Evaluate the integral of e⁻³ˣ from 0 to ∞, or show it diverges.”

Use this for “Show that the integral of 1 ÷ (x² + 9) from 1 to ∞ converges, and find its value” (c = 3).

Use this for “The region between y = 3x and y = x² is turned about the x-axis. Find the volume of the solid.”

Use this for “The region under y = x(3 − x) is turned about the y-axis. Find the volume of the solid by shells.”

Use this for “Find the length of the curve y = 2x³ᐟ² from x = 0 to x = 1.”

Use this for “A force of 30 N holds a spring stretched 0.15 m past its natural length. How much work stretches it from 0.15 m to 0.4 m?”

Use this for “A round tank 1.5 m in radius and 3 m deep is full of water. How much work pumps it all out through a pipe 0.5 m above the rim?”

Use this for “Does Σ n(0.5)ⁿ converge? Use the ratio test, and find the sum if there is one.”

Use this for “Add the first 9 terms of 1 − 1/2 + 1/3 − 1/4 + …. How far from the sum can that be?”

Use this for “Does Σ from n = 1 to ∞ of 1 ÷ (n² + 4) converge? Name the test.”

Use this for “Estimate cos 0.8 with a fourth-degree Maclaurin polynomial and bound the error.”

Use this for “g(2) = 5, g′(2) = 3, g″(2) = −2 and g‴(2) = 12. Use P₃ about x = 2 to estimate g(1.9).”

Use this for “Find the interval of convergence of Σ (x − 1)ⁿ ÷ (4ⁿ√n)” (√n is n0.5).

Use this for “Estimate ∫ from 0 to 0.6 of e−x² dx with three terms of a series, to within how much?”

Use this for “Find the length of one arch of x = 2(t − sin t), y = 2(1 − cos t).”

Use this for “Find the area inside the cardioid r = 2 + 2 cos θ.”

Use this for “Find the slope of the tangent to r = 1 + cos θ at θ = π/2.”

Use this for “A rectangular gate 2 m wide and 3 m tall stands upright with its top 1 m under the water. Find the force on it and where it acts.”

Use this for “The region under y = x² from 0 to 3 turns about the x-axis. Use Pappus’s theorem to find the volume.”

Use this for “How many panels make the trapezoid estimate of ∫ from 1 to 2 of 1/x dx good to 0.0001? How many for Simpson’s rule?”