8 Find f if 2 2 8 f x x x and f2 7 Integration by Substitution Definite

8 find f if 2 2 8 f x x x and f2 7 integration by

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8. Find f if 2 ( ) 2 8 f x x x   and f(2) = 7.
Integration by Substitution – Definite Integrals Change 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 b b a g a f g x dx f u du Steps for Integrating by Substitution—Definite Integrals 1. Choose a substitution u = g(x), such as the inner part of a composite function. 2. Compute ( ) du g x 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. 4 2 2 3 2 8 x x dx 10. 2 3 2 0 4 x x dx 11. 0 sin cos2 x x dx 12. 4 4 cos xdx 13. 2 3 2 x dx AP Calculus Name:____________________________________ Lesson- Areas and Volumes Date:_____________________________________ Area Problems 1. The area of the region bounded between 223yxxand 445yx2. The area of the region bounded bylnyx, the xand yaxes, and 3 . y a. With respect to x.b. With respect to y .
3. The area of the region in quadrant I bounded by228xy, 212xy, and 0 y y . Volume Problems 4. 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
y 8. The volume of the solid generated by rotating the region in quadrant I bounded by21yx, 1xand10yaround   .

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