Integration by Substitution – Definite IntegralsChange of Variables for Definite Integrals: If the function u = g(x) has a continuous derivative on [a, b] and f is continuous on the range of g, then( )()(( ))( )g bbag af g xdxf u duSteps for Integrating by Substitution—Definite Integrals1. Choose a substitution u = g(x), such as the inner part of a composite function.2.Compute ( )dugx dx. Compute new u-limits of integration g(a) and g(b).3.Re-write the integral in terms of u and du, with the u-limits of integration.4.Find the resulting integral in terms of u.5.Evaluate using the u-limits. No need to switch back to x’s!9.422328xxdx10.23204xx dx11.0sincos2xx dx12.44cosxdx13.232x dxAP CalculusName:____________________________________Lesson- Areas and VolumesDate:_____________________________________Area Problems1. The area of the region bounded between 223yxxand 445yx2. The area of the region bounded bylnyx, the xand yaxes, and 3.ya. With respect to x.b. With respect to y.
3. The area of the region in quadrant I bounded by228xy, 212xy, and 0yy.Volume Problems4. Find the volume of the solid whose base is the region in quadrant I that is bounded by3yx, 0y, and 2x. All cross-sections perpendicular to the x-axis are and
y8. The volume of the solid generated by rotating the region in quadrant I bounded by21yx, 1xand10yaround .