Unit 1

Volume Problems

100

Let R be the region bounded by the graphs of y=e^(2x), y=2, and the y-axis. Which integral correctly computes the volume of the solid formed when R is rotated around the line y=2?

What is pi times the integral from 0 to (ln2)/2 of (2-e^(2x))^2dx?

200

This is the integral that correctly computes the volume of the solid formed when R is rotated around the x-axis when R is the region bounded by the graphs of y=sin(x^2) and the x-axis on the interval [0,(pi)^(1/2)].

What is pi times the integral from 0 to the square root of pi of sin^2(x^2)dx.

300

Which of the following integrals gives the volume of the solid whose base is enclosed by the circle x^2 + y^2 = 1 and whose cross sections, taken perpendicular to the base and parallel to the y-axis, are squares?

What is the integral from -1 to 1 of 4(1-x^2)dx?

400

Let R be the region in the first quadrant bounded by the graphs of y=e^(2x), y=2, and the y-axis. Set up, but DO NOT evaluate or simplify, the integral that gives the volume of the solid obtained by rotating the region R around the line x=(ln2)/2 using the DISK/WASHER method.

What is the integral from 1 to 2 of pi[((ln2)/2)^2 - ((ln2)/2 - (lny)/2)2]dy?

Arc Length

100

Find a curve through the point (0,1) whose length integral is L = the integral from 1 to 2 of (1 + 1/y^4)^(1/2) dy.

What is x = -1/y + 1?

200

Find the equation of a curve that passes through the point (1,e) and has an arc length on the interval [0.5,5] given by the integral from 0.5 to 5 of (1+(2/x^4))^(1/2)dx.

What is y=e + 2^(1/2) - (2^(1/2))/x

300

Which integral represents the arc length of the curve y=-lnx on the interval 1 less than or equal to x less than or equal to e^(-2)?

What is the integral from 0 to 2 of (1+e^(-2y))^(1/2)dy?

400

Find the length of the curve y=(1/3)(x^2+2)^(3/2) for 0 less than or equal to x less than or equal to 1.

What is 4/3

Logarithmic/Exponential

100

Find d/dx [5^(3x) + sin(lnx)]

What is (3ln5)5^(3x) + (cos(lnx))/x?

200

Express 3ln(cosx) - ln(x^2 + 1) + (ln4)log(base 4)(3x) as a natural logerithm of a single quantity.

What is ln((3xcos^3(x))/(x^2 + 1)?

300

Find the derivative of y with respect to x for y = (cotx)^(cotx).

What is y = -(cotx)^(cotx)(csc^2(x))(lncotx + 1)?

400

Find d/dx[ln(ln(x^2 + 3))]

What is (2x)/((x^2+3)ln(x^2 +3))?

Inverse Trig

100

Evaluate the integral from -4 to 4 of (3/(4x^2 + 1))dx.

What is (3/2)tan^-1|8| - (3/2)tan^-1|-8|?

200

Simplify sec(sin^-1(x))

What is 1/(1-x^2)^(1/2)

300

Evaluate the integral of (2 - x)/ (1-9x^2)^(1/2) dx.

What is (2/3)(1-9x^2)^(1/2) + (1/9)sin^-1(3x) + C

400

Find the equation of the line tangent to f(x) = cos^(-1)(x/2) at the point (3^(1/2), pi/6).

What is y = -x + 3^(1/2) + pi/6?