Use x_(k)^(**) as the right endpoint of each subinterval to find the area under the curve y=2-(x)/(2) over the interval [-1,4]. Evaluate the integral using integration by parts with the indicated choice of u and v int(e^(2theta)sin(3theta)) d theta. Write out the form of the partial fraction decomposition of the function and evaluate the integral int((3x^(2)-x+8)/(x^(3)+4x)) dx. Evaluate the following integral by using the Gamma function int_(0)^((pi)/(4))sin^(2)(4theta)cos^(3)(2theta) d theta. The curve x=(2/3)y^(3/2), 0 <= y <= 3, find the area of the surface obtained by rotating this curve about the y-axis.