Questions from Larson Chapter 4 Topics. 5. Evaluate

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1 Math. Questions from Larson Chapter 4 Topics I. Antiderivatives. Evaluate the following integrals. (a) x dx (4x 7) dx (x )(x + x ) dx x. A projectile is launched vertically with an initial velocity of 6 feet per second from a height of feet above the ground. (a) Find the function s(t) that provides the height of the projectile t seconds after it is launched. Use the result of (a) to determine the maximum height reached by the projectile.. A projectile is launched from a height of 7 feet above the ground. Find the initial velocity required for the projectile to reach a maximum height of 8 feet above the ground. 4. Suppose dy dx = sin x + sec x and y() =. Find y.. Evaluate (cos t + sin t + ) 6t dt 6. An airplane taking off from a runway travels feet before lifting off. The airplane starts from rest, moves with constant acceleration, and makes the run in 4 seconds. With what speed does it lift off? 7. A stone is thrown straight down from the edge of a roof, 64 feet above the ground, at a speed of 8 feet per second. (a) Find s(t) the height of the stone in feet, at time t, for t until the stone hits the ground. What is the height of the stone seconds after it was released? How many seconds after the stone was released does it hit the ground? (d) What is the impact velocity of the stone? 8. Suppose f (x) = 8x 4, f () = and f() = 8. Find: (a) f(x), and f(). t

2 II. Area and Riemann Integrals. (a) Use the definition of area to find the area bounded by y = 64 x, the x-axis and the lines x = and x =. Use the definition of area to find the area bounded by y = 64 x, the x-axis and the lines x = and x = 4. Use your answer in (a) and to find the area bounded by y = 64 x, the x-axis and the lines x = and x = 4.. Describe the region whose area is given by the 7 integral 49 x dx and use geometry to 7 determine the value of the integral.. (a) Use geometry to evaluate the integrals 9 x dx and x dx. Use your results from (a) to evaluate the integral Given that and (4 9 x x )dx. g(x)dx = find f(x)dx = 4, (f(x) 4g(x))dx. f(x)dx = 7. Consider the piecewise defined function { x if x ; f(x) = x if < x. Use geometry find f(x) dx. 4. Suppose f and g are integrable on the closed intervals determined by, and. Given that (a) (d) f(x)dx = 8, = 7, evaluate: (e ) (f) f(x) dx and [f(x) + g(x)] dx and f(x) dx and f(x)dx =, and 8f(x) dx [8f(x) g(x)] dx and [8f(x) + g(x)] dx [f(x) g(x)] dx [g(x) + f(x)] dx + [f(x) g(x)] dx (g) If α and β are two constants, write (αf(x) + βg(x)) dx in terms of α and β (h) + 6. Interpret the limit of sums lim n n i= ( 4 + i n ) / ( ) n as an integral, and describe the region whose area it represents (you do not need to find the area). 7. Use the a limit process to find area of the region that lies between the x-axis and the curve y = 9 x for x.

3 III. Fundamental Theorem of Calculus. (a) Let F (x) = Let G(x) = x 4 x x t t sin t dt. Find F (x). t t sin t dt. Find G (x). 4. Evaluate π/ [ cos x + sin x tan x sec x 6x ] dx.. Consider a continuous function such that f(x) = π and 6 f(x)dx =. (a) What is the average value of f for x 6. If f is extended to be even, evaluate f(x)dx. If f is extended to be odd, evaluate f(x)dx. (d) Evaluate 6 (f(x) + )dx.. Find the area of the region that lies below y = x + x 64 and above the x-axis. 6. Use the Fundamental Theorem of Calculus to find the derivative of F (x) = 4 x tan(t )dt 7. Use the Fundamental Theorem of Calculus to find the derivative of h(x) = sin(x) (cos(t 4 ) + t)dt. Find the area under the curve y = sin x+4x for x π. ( + x x ) dx (if it ex- 8. Evaluate the integral ists).

4 IV. Integration by Substitution. Evaluate cos t dt Hint: cos t =. Evaluate: 9r dr (a) r + cos t x x dx x π + x 4 dx Hint: for use symmetry to deduce the answer, do not try to find an antiderivative.. (a) Suppose f (x) = (x + ) x, find f. Evaluate (x + ) x dx 4. Find the integral x(x + ) 4 dx.. Find (7x ) sec(7x 4x) tan(7x 4x) dx. Suppose that f is continuous on the interval [, ]. (a) Given that f(x) dx =. Find: / / /7 /7. Evaluate f(x ) dx xf(x ) dx f(7x) dx 4 f(x) dx = and x x dx. If the integral does not exist, state why it doesn t exist.. Find an equation for the function f(x) that has the given derivative and initial value. f (x) = x(x + ) f( ) = 4. Evaluate 7 t dt using u- + 8t + 6 substitution and the Fundamental Theorem of Calculus. 6. Find 7. Find (x + 4) csc(x + x) cot(x + x) dx 4x sec (x + ) dx 4. Suppose that. f( 8t) dt. f(t) dt =. Calculate. Using the method of u-substitution, evaluate π 8 sin(6x) cos 6 (6x) dx. 8. Given that e e e x + x dx 4x dx = 4 find + x 9. Find the area of the region that lies below the graph of f(x) = (4x ) x, above the x- axis and between the vertical lines x = and x =. 6. Using the method of u-substitution, write 4 where u = du = a = b = f(u) = x ( + x 4 ) dx = b a f(u) du The value of the original integral is

5 V. Numerical Integration. (a) Estimate sin(x )dx with n = 6 subintervals using the trapezoidal rule. Estimate sin(x )dx with n = 6 subintervals using Simpson s rule. What is the largest error you would expect in evaluating rule with n =, subintervals? x dx using Simpson s (d) What is the largest error you would expect in evaluating x dx using the trapezoidal rule with n =, subintervals? (e) Find n so that the error in approximating 8 dx using the trapezoidal rule does not exceed x.. (f) Find n so that the error in approximating 8 dx using Simpson s rule does not exceed x... Suppose over a second interval, the speed of the Walla Walla University Garbage truck is measured every two seconds with speeds v() = ft/s, v() = ft/s, v(4) = ft/s, v(6) = ft/s, v(8) = 7 ft/s and v() = ft/s. Use the trapezoidal rule to estimate how many feet the truck travelled in the second interval.. (a) Use the trapezoidal rule with n = 6 to estimate π sin x dx. Use the trapezoidal error formula to determine the largest possible error you would expect in your answer to (a) (you will be given the error formulas for Simpson s Rule and the Trapezoidal Rule if needed on the final test)? 4. Approximate the integral e x dx using n = 4 and the following methods: (a) Trapezoidal Rule, Midpoint Rule, Simpson s Rule. Determine a smallest n so that Simpson s rule will approximate the integral 6 dx + x with an error E n satisfying E n Determine a smallest n so that the trapezoidal rule will approximate the integral 8 6 dx 7 + x with an error E n satisfying E n.. 7. Estimate n = 4. 4 x dx using Simpson s Rule with 8. The width w n, in feet, at various points x n along the foot-long fairway of a golf course changes as shown in the table below. n 4 x n w n n x n w n If one pound of fertilizer covers square feet, find a reasonable estimate for the amount of fertilizer needed to fertilize the fairway using both the trapezoidal rule and Simpson s rule.

Chapter 4 Integration

Chapter 4 Integration SECTION 4.1 Antiderivatives and Indefinite Integration Calculus: Chapter 4 Section 4.1 Antiderivative A function F is an antiderivative of f on an interval I if F '( x) f ( x) for