# Calc Final Review - Problems (Please Explain)

• May 11th 2009, 07:39 PM
Dreiz
Calc Final Review - Problems (Please Explain)
3. Evaluate the integral http://upload.wikimedia.org/math/f/e...eb485cd0f6.pngsin^5x cos^3x dx , interval [pi/2, 3pi/4]

14. Evaluate the integral http://upload.wikimedia.org/math/f/e...eb485cd0f6.png (x^3)/sqrt(16-x^2) dx, interval [0, 2sqrt3]

19. Evaluate the integral http://upload.wikimedia.org/math/f/e...eb485cd0f6.png1/(x^2-1) dx, interval [2, 3]

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6. Use midpoint rule and simpsons rule to approximate the given integral with the specified value of n.

8. Use the trapezoidal rule and the midpoint rule and the simpsons rule to approximate the given integral with the specified vaule of n.
http://upload.wikimedia.org/math/f/e...eb485cd0f6.png sin(x^2)dx, interval [0, 1/2] n=4

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Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

6. http://upload.wikimedia.org/math/f/e...eb485cd0f6.png 1/(2x-5) dx, interval [0, -infinity]

20. http://upload.wikimedia.org/math/f/e...eb485cd0f6.png lnx/x^3 dx, interval [1, infinity]

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4. Functions given x=y^2 - 4y and x=2y - y^2 points of cross over (-3,3) and (0,0). Find the area of which curves overlap.

14. Sketch the region enclosed by the given curves. Decide whether to intergrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the are of the region.
y=singx, y=2x/pi , x is greater of equal to 0

38. Find the area of the region bounded by the parabola y=x^2, the tangent line to this parabola at (1,1) and the x-axis.

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8. Find the volume of the solid obtained by rotating the region bounded by the given curvbes about the specified line. Sketch the region, the solid, and a typical disk or washer.
y=x^(2/3), x=1, y=0, about the y-axis

22. Each integral represents the volume of a solid. Describe the solid.
a. pi http://upload.wikimedia.org/math/f/e...eb485cd0f6.pngy dy, interval [2,5] b. pi http://upload.wikimedia.org/math/f/e...eb485cd0f6.png [(1+cosx)^2 - 1^2]dx, interval [0,pi/2]

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1. Use the arc length formula (2) to find the length of the curve y=2-3x, interval [-2,1]

formula L = http://upload.wikimedia.org/math/f/e...eb485cd0f6.png sqrt [ 1+ (dy/dx)^2 ] dx interval [a,b]

8. Graph teh curve and find its exact length.
y=(x^2)/2 - (lnx)/4, interval from 2 to 4

20. Use either a CAS or a table of integrals to find the exact length of the curve.
y = lnx interval from 1 to sqrt(3)

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Please explain how to do these problems, no need for solutions but if you have some spare time it would be helpful.
Thank you
• May 11th 2009, 08:28 PM
derfleurer
3.) $\int sin^4xcos^3xdx = \int sin^4x(1 - sin^2x)cosxdx = \int (sin^4xcosx - sin^6xcosx)dx$

14.) $\int \frac{x^3}{\sqrt{16 - x^2}}dx = \int \frac{(4sin\theta)^3}{4cos\theta}4cos\theta d\theta = 64\int sin^3\theta d\theta$

$x = 4sin\theta$

$dx = 4cos\theta d\theta$

19.) $\int \frac{1}{x^2 - 1}dx = \int \frac{1}{(x - 1)(x + 1)}dx$

$\frac{A}{x - 1} + \frac{B}{x + 1} = \frac{1}{x^2 - 1}$

$A(x + 1) + B(x - 1) = 1$

$A = 1/2 ... B = -1/2$

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4.) $\int_{0}^{3} [(2x - x^2 + 4) - (x^2 - 4x + 4)]dx$

38.) $f(x) = x^2$

$1 = f'(1)(1) + b$

$g(x) = 2x - 1$

$f(x) = g(x)$ @ $x = 1$

$\int_{0}^{1} [x^2 - (2x - 1)]dx$

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8.) $f(x)^{2/3}$

$\int_{0}^{1} (1 - y^{3/2})dy$

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1.) $\int_{a}^{b} \sqrt{1 + (\frac{dy}{dx})^2}dx$

$y = 2 - 3x$

$\frac{dy}{dx} = -3$

$\int_{-2}^{1} \sqrt{1 + (-3)^2}dx$

8.) $y = \frac{x^2}{2} - \frac{lnx}{4}$

$\frac{dy}{dx} = x - \frac{1}{4x}$

$\int_{2}^{4} \sqrt{1 + (x - \frac{1}{4x})^2}dx$
• May 12th 2009, 02:07 AM
mr fantastic
This thread is going to become an absolute shambles if replies and counter replies are posted to all these questions. Furthermore, the use of identical numbering (despite the ---------------------) is going to create a lot of problems.

@OP: Work out what questions haven't been replied to and re-post them in new threads if you still need help. No more than three questions per thread. Note also rule #1 here: http://www.mathhelpforum.com/math-he...php?do=cfrules, so please don't re-post any questions that have already been given a reply.