- 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 IntegralsKnow how to use the Comparison Test to determine if an improper integral conv./div. - ex. Does 3121xdxxx∞++∫converge? Notice 32111~21xdxdxxxx∞∞++∫∫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 greaterthan the unknown function and convergesif you suspect the function converges You have to pick a function that is smallerthan the unknown function and divergesif you suspect the function diverges- ex. Suppose 10( )f x dx∫converges and h(x) < f(x) < g(x) for 0 §x§1. Does 10( )g x dx∫converge? Does 10( )h x dx∫converge? Explain. Know the following integrals for comparison - 11pxdx∞∫converges for p> 1 and diverges for p≤1. - 101pxdx∫converges for p< 1 and diverges for p≥1. - 0axedx∞−∫converges for a> 0.
Math 10B Final Exam Review Outline10 Sections 8.1 / 8.2: Areas and Volumes / Applications to GeometryKnow 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 2baVolumeR dxπ=∫, where Ris the radius - ex. Find the volume of the region revolved around the x-axis and bounded by the curves y= x3, the x-axis, x= 0 and x= 2. - ex. A solid is formed by rotating the region bounded by y= 6x– x2and 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= pis 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 yx=, 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 )yx=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 ()22baVolumeRrdxπ=∫−, where Ris the outer radius and ris the inner radius - ex. Sketch the region bounded by the parabola y= x2and the line y= 3x. 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 22()aaaxdxπ−−∫represents the volume of a sphere of radius a.