Chapter 6 | Applications of Integration
733
10
10 x
339. [T] Find the surface area of the shape created when rotating the curve in the previous exercise from x =1 to x =2 around the x -axis. If you are unable to find intersection points analytically in the following exercises, use a calculator. 340. Find the area of the hyperbolic quarter-circle enclosed by x =2and y =2 above y =1/ x .
330. ∫
t − ∫
dt
dt t
5
5 x
e π
−1
331. ∫
x + ∫
dx
dx x
1
−2
1
dx ∫ x
332. d
dt t
[T] Find the arc length of y =1/ x from
341.
x 2
dx ∫ x
333. d
dt t
x =1 to x =4. 342. Find the area under y =1/ x and above the x -axis from x =1 to x =4. For the following exercises, verify the derivatives and antiderivatives. 343. d dx ln ⎛ ⎝ x + x 2 +1 ⎞ ⎠ = 1 1+ x 2 344. d dx ln ⎛ ⎝ x − a x + a ⎞ ⎠ = 2 a ⎛ ⎝ x 2 − a 2 ⎞ ⎠
334. d
x +tan x )
dx ln(sec
For the following exercises, use the function ln x . If you are unable to find intersection points analytically, use a calculator. 335. Find the area of the region enclosed by x =1 and y =5 above y = ln x . 336. [T] Find the arc length of ln x from x =1 to x =2. 337. Find the area between ln x and the x -axis from x =1 to x =2. 338. Find the volume of the shape created when rotating this curve from x =1 to x =2 around the x -axis, as pictured here.
⎛ ⎝ ⎜ 1+ 1− x 2 x
⎞ ⎠ ⎟ = − 1
345. d
dx ln
x 1− x 2
⎛ ⎝ x + x 2 − a 2 ⎞
346. d
⎠ = 1
dx ln
x 2 − a 2
347. ∫
dx x ln( x )ln(ln x ) = ln ⎛
⎝ ln(ln x ) ⎞
⎠ + C
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