Finding volume
Let R be the "triangular" region in the first quadrant that is bounded above by the line y = 1, below by the curve y = ln x, and on the left by the line x = 1.
Find the volume of the solid generated by revolving R about
a. the x-axis.
Finding arc length
Find the length of the curve
y = ∫ from 0 to x of √(cos(2t)) dt, 0 ≤ x ≤ π/4.
18. Finding volume (Continuation of Exercise 17.) Find the volume of the solid generated by revolving the region R about:
a. the y-axis.
For each x > 0, let G(x) = ∫(from 0 to x) e^(-xt) dt. Prove that xG(x) = 1 for each x > 0.
Evaluate the integrals in Exercises 1–6.
∫ (arcsin x)² dx
Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.
∫ cos t dt / (1 - cos t)
