Calculus 2

Test File

Spring 2000

Test #1

For problems #1 - 4, consider the region bounded by f(x) = x², y = 0, x = 2.

1.) Draw a graph showing the region, with the region shaded in.

2.) Suppose the region is revolved around the x-axis. SET UP BUT DO NOT EVALUATE the integral for finding the volume of the resulting solid using cylindrical shells.

3.) Suppose the region is revolved around the x-axis. SET UP BUT DO NOT EVALUATE the integral for finding the volume of the resulting solid using disks.

4.) Using the calculator to evaluate any necessary integrals, find the perimeter of the region.

5.) Suppose the curve y = x², 0 £ x £ 2, is revolved around the line y = 6. SET UP the integral for finding the surface area and use your calculator to evaluate it.

For problems number 6 and 7, consider the region bounded by the graphs of y = 2x² and x = 2y².

6.) Find the volume of the solid when that region is revolved around the x-axis. Exact Answer.

7.) Find the volume of the solid when that region is revolved around the line y = 3.. Exact Answer.

8.) Consider a square based pyramid, with base area, A, and height, h. Use cross-sections and integration to find the volume.

Test #2

Use the sheet of integral formulas to work these two problems. Write the number of any formula you use. When you are done, hand in this sheet AND the integral formulas. Then get the rest of the test.

1.)

2.)

Test #2 - Part 2

1.) ò ln x dx 2.) ò sin³x cos²x dx

3.) ò x³ sin 2x dx 4.) ò tan³x sec³x dx

5.)

6.) ò sec³x dx

7.) A conical tank is resting on its base, which is at ground level, and its axis is vertical. The tank has radius 5 ft and height 10 ft and is full of water (density = 62.4 lb/ft³). Compute the work done in emptying the tank by pumping the water out a pipe that extends 5 feet out the top of the tank.

8.) A spring has a natural length of 1 m, and a force of 10 N is required to hold it stretched to a total length of 2 m. How much work is done in compressing this spring from its natural length to a length of 60 cm?

Test #3

1.)

2.) EXACT ANSWER!!

3.) EXACT ANSWER!!

4.) EXACT ANSWER!!

5.)

6.) Find the volume if the region from #5 is revolved around the y-axis.

7.)

8.)

9.) The region bounded by the hyperbola x² - 4y² = 1 and the line x = 2 is revolved around the x-axis. Find the volume of the resulting solid.

10.) Find the equation of the ellipse where the point, (x, y), is on the ellipse if and only if its distances from the points (2, 0) and (8, 0) add to 8.

Test #4

1.) Find the area inside ONE loop of r = cos 4q.

2.) a.) Convert (-3, 3) to polar coordinates.

b.) Convert (3, 5p/6) to rectangular coordinates.

3.) SET UP AND USE YOUR CALCULATOR TO EVALUATE the integral necessary for finding the surface area of the surface formed by rotating the curve given by x = e^2t, y = sin t, 0 £ t £ p, around the x-axis.

4.) Find the arc length given by x = t cos t, y = t sin t, 0 £ t £ p/6. Give the EXACT ANSWER.

5.) Graph r = e^q.

6.) Convert r = cos q + sin q to rectangular coordinates and graph it.

7.) Find the area inside BOTH r = 2 and r = 3 + 2 cos q. Give the EXACT ANSWER.

8.) Set up and USE YOUR CALCULATOR TO EVALUATE the integral for finding the perimeter of one loop of r = cos 3q.

9.) Write the parametric function x = cos t, y = 1 - sin t, 0 £ t £ 2p, in terms of just x and y. Graph it and show the starting point, direction of motion and ending point.

Test #5

1.) Graph the following conic and fill in the blanks. For any blank that is not applicable to the particular conic, write "DNA."

y - 4x² + 16x = 0

Center Vertex or Vertices____________

Focus or Foci Directrix______

Asymptote(s)______________________

End pts. Minor Axis___________________

2.) Graph the following conic and fill in the blanks. For any blank that is not applicable to the particular conic, write "DNA."

Center Vertex or vertices____________

Focus or Foci____________

Asymptote(s)______________

End pts. Minor Axis___________________

3.) Graph the conic 3x² - 6x - y² + 6y = 0. Fill in all the following blanks. If one does not apply to this particular conic, write "DNA."

Center Vertex or vertices____________

Focus or Foci____________

Asymptote(s)______________

4.) Find an equation for the parabola with focus (2, 3) and vertex (2, 6).

5.) Find an equation for the ellipse with foci (2, 3) and (8, 3) and a vertex at (10, 3).

6.) Find an equation for the hyperbola with asymptotes y = 2x and y = -2x and a vertex at (4, 0).

7.) Determine convergence or divergence for the sequence {a_n} with as a_n given.

a.) a_n = n sin(1/n)

b.) a_n = tan^-1n.

c.) a_n =

d.) a_n =

Test #6

In problems #1 and 2, determine convergence or divergence for the following sequences. Show all necessary work to justify your answer. If the sequence converges, state its limit.

1.) 2.)

3.) Find the third degree Taylor Polynomial for f(x) = sin x, with a = p/4.

In problems #4 - 7, determine convergence or divergence for the following series. Show all necessary work to justify your answer.

4.) 5.) 6.)

7.)

8.) Determine convergence or divergence for the following series. Show all necessary work to justify your answer. If the series converges, find the sum.

9.) Prove that the harmonic series diverges.

10.) Consider the following series.

a.) Find the sum of the first 90 terms.

b.) You may assume the integral test applies. The remainder term for the integral test satisfies the following inequality
. Use the inequality, and your answer from part a.), to find an interval that contains the sum of the entire series.

Test #7

Determine convergence (absolute or conditional) or divergence for each of the following series.

1. 2. 3.

For each of the following power series, find the interval of convergence. Do not forget to check the endpoints.

4. 5.

6.

Do the following.

7. Use a familiar power series to find a power series for the following function. Then find the interval of convergence. Include endpoints.

8. Use the first 6 terms of the power series for sin x to approximate sin(0.1). Then, find an upper bound for the error of this estimate.