Notice that above different variables for limits are used for different

Notice that above different variables for limits are

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- Notice that above different variables for limits are used for different integrals! If any part of the integration is equal to infinity, the integral diverges. - Infinities do not cancel out! (“ - 0”) Section 7.8: Comparison of Improper Integrals Know how to use the Comparison Test to determine if an improper integral conv./div. - ex. Does 3 1 2 1 x dx x x + + converge? Notice 3 2 1 1 1 ~ 2 1 x dx dx x x x + + for large x . - We only look at the largest power in the numerator and denominator - Think of the strategy for finding horizontal asymptotes You have to pick a function that is greater than the unknown function and converges if you suspect the function converges You have to pick a function that is smaller than the unknown function and diverges if you suspect the function diverges - ex. Suppose 1 0 ( ) f x dx converges and h ( x ) < f ( x ) < g ( x ) for 0 § x § 1. Does 1 0 ( ) g x dx converge? Does 1 0 ( ) h x dx converge? Explain. Know the following integrals for comparison - 1 1 p x dx converges for p > 1 and diverges for p 1. - 1 0 1 p x dx converges for p < 1 and diverges for p 1. - 0 ax e dx converges for a > 0.
Math 10B Final Exam Review Outline 10 Sections 8.1 / 8.2: Areas and Volumes / Applications to Geometry Know how to use horizontal/vertical slices to set up a definite integral to find areas Know how to use horizontal/vertical slabs to set up a definite integral to find volumes - ex. See Section 5.4 for some examples of area problems. Know how to calculate volumes of solids of revolution by using slabs perpendicular to the axis of symmetry to set up a definite integral Know how to calculate volumes of solids of revolution that are revolved around the x - axis, the y -axis, any horizontal line, or any vertical line The formula for such problems is 2 b a Volume R dx π = , where R is the radius - ex. Find the volume of the region revolved around the x -axis and bounded by the curves y = x 3 , the x -axis, x = 0 and x = 2. - ex. A solid is formed by rotating the region bounded by y = 6 x x 2 and y = 0 about the x -axis. Find the volume of this solid. - ex. The region bounded by the curves y = sin( x ), y = 0, x = 0 and x = p is rotated about the x -axis. Calculate the resulting volume. - ex. Compute the volume of the solid formed by rotating about the x -axis the region bounded by y x = , y = 1 and the y -axis. - ex. Find the volume of the solid obtained by rotating the region enclosed by the graph of sin(3 ) y x = and the x -axis from x = 0 to x = p /3 about the x -axis. Know how to calculate volumes of solids of revolution that have either a circular slab or a circular slab with a "hole" The formula for such problems is ( ) 2 2 b a Volume R r dx π = , where R is the outer radius and r is the inner radius - ex. Sketch the region bounded by the parabola y = x 2 and the line y = 3 x . Set up an integral to compute the area of the region. (Don’t compute it.) Set up an integral to compute the volume of the solid formed by rotating the region about the x - axis. (Don’t compute it.) - ex. T/F: The integral 2 2 ( ) a a a x dx π represents the volume of a sphere of radius a .