Barrons AP Calculus - David Bock

or f ′(θ) dθ. See Questions 14, 15, and 44 in the Practice Exercises. BC ONLY Region Bounded by Polar Curve FIGURE N7–6 To find ...

we see that the desired area is twice the sum of two parts: the area of the circle swept out by θ as it varies from 0 to plus th ...

B. VOLUME B1. Solids with Known Cross Sections If the area of a cross section of a solid is known and can be expressed in terms ...

ΔV = (2y)^2 Δx = 4y^2 Δx = 4(9 − x^2 ) Δx. Now, using symmetry across the y-axis, we find the volume of the solid: Questions 25, ...

FIGURE N7–12 The volume ΔV of a typical disk is given by ΔV = π y^2 Δx. The equation of the circle is x^2 + y^2 = r^2. To find t ...

disks. WASHERS A washer is a disk with a hole in it. The volume may be regarded as the difference in the volumes of two concentr ...

FIGURE N7–15 EXAMPLE 9 Find the volume of the solid generated when the region bounded by y = x^2 , x = 2, and y = 0 is rotated a ...

A cylindrical shell may be regarded as the outer skin of a cylinder. Its volume is the volume of the rectangular solid formed wh ...

(Note that we obtained the same result using disks in Example 7.) FIGURE N7–18 EXAMPLE 11 The region bounded by y = 3x − x^2 and ...

Shell. (Note that we obtained the same result using washers in Example 8.) FIGURE N7–20 NOTE: In Examples 32 and 33 we consider ...

BC ONLY The parenthetical clause above is equivalent to the requirement that the curve is traced out just once as t varies from ...

EXAMPLE 16 Find the length of the arc of y = ln sec x from x = 0 to SOLUTION: D. IMPROPER INTEGRALS There are two classes of imp ...

Improper integrals of class (1), where the interval is not bounded, are handled as limits: where f is continuous on [a,b]. If th ...

FIGURE N7–23 EXAMPLE 19 EXAMPLE 20 BC ONLY EXAMPLE 21 EXAMPLE 22 Thus, this improper integral diverges. EXAMPLE 23 Since this li ...

Note, however, that it does not become infinite; rather, it diverges by oscillation. Improper integrals of class (2), where the ...

It can be shown that (a > 0) converges if p < 1 but diverges if p 1. Figure N7–25 shows an interpretation of in terms of a ...

Neither limit exists; the integral diverges. NOTE: This example demonstrates how careful one must be to notice a discontinuity a ...

Since converges and dx converges by the Comparison Test. EXAMPLE 31 Show that converges. SOLUTION: we will use the Comparison Te ...

FIGURE N7–27 SOLUTION: About the x-axis. Disk. BC ONLY EXAMPLE 33‡ Find the volume, if it exists, of the solid generated by rota ...

FIGURE N7–28 SOLUTION: About the y-axis. Shell. ΔV = 2πxy Δx = 2π Δx. Note that diverges to infinity. Chapter Summary In this ch ...