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10:07
6:04 6:04Finding 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.
2. When applying the formula for integration by parts, how do you choose the u and dv? How can you apply integration by parts to an integral of the form ∫ f(x) dx?
Finding volume
The region in the first quadrant enclosed by the coordinate axes, the curve y = e^x, and the line x = 1 is revolved about the y-axis to generate a solid. Find the volume of the solid.
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (z + 1) / [z²(z² + 4)] dz
Use the substitution z = tan(θ/2) to evaluate the integrals in Exercises 41 and 42.
∫ csc θ dθ
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(x + 1) / (x² (x − 1))] dx
