Study Guide for Test 2

Transcription

1 Study Guide for Test Math 6: Calculus October, 7. Overview Non-graphing calculators will be allowed. You will need to know the following:. Set Pieces Trigonometric Substitutions (Section 7.).. Partial Fractions (Section 7.4). 4. Approximate Integrals (Sections 7.7). 5. Improper Integrals (Section 7.8) 6. Arc Length of a Curve. (Section 8.). 7. Surface Area of a Surface of Revolution. (Section 8.). 8. Integration Practice. (Section 7.5, etc.).. Set Pieces Set Piece 9 Problem. Find the area in the ellipse defined by the equation ( x ) ( y ) + =. Solution. Step : Sketch. The x-intercepts are (, ) and (, ). The y-intercepts are (, ) and (, ). Here is a sketch of the ellipse: [Insert sketch here] We will concentrate on finding the area in the first quadrant. This will give one-fourth of the area of the ellipse.

2 Step : Set-up the area integral. We need to express the y-coordinate of the ellipse as a function of x. So we solve for y: ( y ) ( x ) x = = 9, so y = x 9 9 = 9 x 9 = 9 x, thus y = 9 x. So the area under this curve, in the first quadrant, is 9 x dx = 9 x dx. Step : Set up a right triangle and convert to a trig functions. We want to use the Pythagorean Theorem to help give formulas for x, dx, and 9 x. To do this, set z = 9 x. So z = 9 x. Hence, x + z =. This gives us a right triangle which we draw below: [ Insert right triangle here. Label an angle θ, ] [ label the adjacent leg z = 9 x, ] [ label the opposite leg x, ] [ and label the hypotenuse. ] To find a good substitution formula for x, we use the formula sin θ = opp hyp = x. So Thus x = sin θ. dx = cos θ dθ. Now to find a formula for 9 x we use the formula cos θ = adj hyp = 9 x. So 9 x = cos θ. Step 4: Convert the integral. First we have to worry about the limits of the integral. The lower limit is x =. Since x = sin θ we have = sin θ. So sin θ =. The angle that does this is θ =. So we replace x = with θ = as the lower limit. The upper limit is x =. Since x = sin θ we have = sin θ. So sin θ =. The angle that does this is θ = π. So we replace x = with θ = π as the upper limit. We now use the above formulas to convert the integral: 9 x dx = ( )( ) cos θ cos θ dθ = 9 cos θ dθ = 6 cos θ dθ. Other angles do this as well: for example θ = π. However, we want to stick to angles in the first quadrant: we want angles between and π/. This is all we need to yield all the x-values (uniquely) between and.

3 Step 5: Solve the integral. We use the formula cos θ = ( + cos θ). So the integral becomes 6 cos θ dθ = 6 ( + cos θ) dθ = We simplify by using u = θ. Thus du = dθ and dθ = du. So ( + cos θ) dθ = ( + cos u) du = ( + cos θ) dθ. ( + cos u) du. Observe that the upper limit θ = π/ becomes u = π since u = θ. Now we evaluate the integral: π ( + cos u) du = [ ] π u + sin u = [ ] π + sin π [ ] + sin = [ ] π + [ ] + = π. Step 6: multiple by four. Now we take the answer and multiply by four. We need to do this since the integral from Step is one fourth the desired area. So the final answer is 6π. Area = 4 π = 6π. Set Piece Problem. Find the volume obtained by rotating the region under y = x = about the y-axis. Comment. This uses the shell method and partial fractions. x +x+ between x = and Set Piece Torricelli s infinitely long solid. Rotate the region under the curve y = /x with x about the x-axis. Problem.. Sketch this infinitely long solid. Problem.. Show that it has finite volume and find its exact volume. Note: we also learned that the surface area is infinite. It is interesting that a solid with infinite surface area can have finite volume. Set Piece Derive the formula A = 4πR for the surface area of the sphere or radius R. Comment. This uses the surface area formula da = πrds. The sphere is thought of as a surface of rotation about the y-axis. The integral goes from to R.

4 Set Piece Derive the formula V = 4 πr for the volume of the sphere of radius R. Comment. This uses the shell method and the formula dv = πrhdx. The sphere is thought of as a volume of rotation about the y-axis. The integral goes from to R. Set Piece 4 Problem. Prove the following: The integral converges if p >, but diverges if p. Comment. The proof divides into three cases: p >, p =, p <. Trigonometric Substitution (Section 7.) Use the Pythagorean theorem to generate a reference triangle. Know all your trig functions. There are three cases as discussed on the chart page 49. Look at Examples and 4 on page 49, and Set Piece 9 in this study guide. 7.:,, 5,,, 5, 7 Partial Fractions (Section 7.4) Learn to do Case I on page Learn to do Case II on page 499. Don t worry about Case III on page 5. If the degree of the numerator is greater than or equal than the degree of the denominator, then don t forget to divide and use the technique on the remainder. Study Example on page 496. Study Example on page 497 (but in class we used the method discussed in the Note on page 498). Study Examples, 4 on page Also study Set Piece. 7.4: 7, 9,, 5, 7, 9, 4 x p Approximate Integrals (Sections 7.7) You will be expected to know three methods for approximating integrals: (i) Simpson s rule, (ii) the trapezoid rule, (iii) the midpoint rule. Study Examples, 4, and 5 on pages Don t worry about the error bounds. But know that Simpson s method is usually the best (for the same n). 7.7: 7, 9,,, 9, 4

5 4 Improper Integrals (Section 7.8) Know all types of improper integrals in the box on page 5. See Examples,,, 5 on pages 5 to 54. Review L Hospital s rule (Section 4.4). See Example 8 on page 56. Know the p test on page 5 (Example 4). The proof of this in one of our set pieces. Know the comparison theorem on page 56, and how to use it (for example in Problem 5 in Section 8.. See also Example on page 57). 7.8:, 5, 7, 9,,, 7 5 Arc Length. (Section 8.) Know how to compute ds using the formula ( ) dy ds = + dx or ds = + dx ( ) dx dy. dy Use ds to find arc length using the formula S = b a ds. Oftentimes you will only have to set up the integral. Do not worry about the arc length function on page :,, 7, 7, 8, 9,, 6 Area of a Surface of Revolution. (Section 8.) Use the formula A = b a πrds to find surface areas of a surface of revolution. For example, Set Piece. The hardest part is computing ds. Just use the same formula as in Section 8.. The formula for r is as follows: r = y if the axis is the x-axis, and r = x if the axis is the y-axis. If the axis is something else, you need to figure r out from the situation (but that won t happen in this test). Oftentimes you will only have to set up the integral. 8.:,, 5, 7, 7, 5, 9 In problem 5, be sure and use the comparison test. 7 Integration Practice 7.5:, 7, 9,, 5, 9 5