Definite Integrals Questions and Answers

A cylindrical can is being filled at the rate of dh/dt = 3t², h(t) describes the height of the contents. Use an integral to write a formula which describes how much the height of the contents have changed between [1, 4]? Do not evaluate the integral.

R is the region bounded by the functions f(x) = 10/x and g(x) = x/10. Find the area of the region bounded by the functions on the interval [7, 10]. Enter an exact answer. Provide your answer below: A = ______ units^2.

Find the derivative, g'(x), of g(x) = ∫ cost dt using two different methods as directed: (a) Find the definite integral for g(x) using FTC2, then take the derivative to find g'(x). (b) Find g'(x) using FTC1.

Assume v(t) = 5t - 15 on the interval [0, 6]. Find the total distance that the object traveled on [0, 6]. An object moves along a line, where its velocity function is as described by the graph.

If an object is moving along a linear path with v(t) = 3t² + 3t - 5. At time t =0 the position of the object is 6. Write the formula that describes the object's position at time t = 8. Do not evaluate the integral.

R is the region bounded by the functions f(x) = (5x/2)-2 and g(x) = x - 8 over the interval [a, b] where a = 1 and b = 4. Represent the area A of R by writing an integral with respect to x. Yo do not need to simplify. Provide your answer below:

R is the region bounded by the functions f(x) =11/(-1 + x) and g(x) = x – 11 . Find the area of the region bounded by the functions on the interval [8, 12).

How many solutions exist for the equation cos 2θ - sinθ = 0 on the interval [0, 360º)? O 0 O 1 O 2 O 3

List the potential rational zeros of the polynomial f(x) = 3x⁵ – x²+2x+18.

R is the region bounded by the functions f(x) = 3 - 2 sin (x) and g(x) = cos (x). Find the area of the region bounded by the functions on the interval [0, π/2]. Enter an exact answer. Provide your answer below: A = ______units^2.

R is the region bounded by the functions f(x) = 2 - 3 cos (x) and g(x) = 3 sin (x). Find the area of the region bounded by the functions on the interval [0, π/2]. Enter an exact answer. Provide your answer below: A = _____units^2.

R is the region bounded by the functions f(x) = 11/-1+x and g(x) = x - 11. Find the area of the bounded by the functions on the interval [8, 12]. A = ______ units²

R is the region bounded by the functions f(x) = 5√(x-1) and g(x) = (5x/3) + (7/3). Find the area A of R. Enter an exact answer. Provide your answer below: A = ____________units^2.

R is the region bounded by the functions f(x) = -(3x/2) - 10 and g(x) = (x/2) + 6 over the interval [a, b] where a = -5 and b = -2. Represent the area A of R by writing an integral with respect to x. You do not need to simplify. Provide your answer below:

Analyze and then sketch the function 3x/(x^2 + 7x + 2). > Determine the interval of increasing/decreasing and the concavity. SHOW YOUR WORK | 3x^4 - 17^3 + 16 = 7x^2 > Determine the interval of increasing/decreasing and the concavity.

R is the region bounded by the functions f(x) = -3x – 3 and g(x) = - (9x/4)+ 3 over the interval [a, b] where a = -2 and b = 1. Represent the area A of R by writing an integral with respect to x. You do not need to simplify. Provide your answer below:

R is the region bounded by the functions f(x) = 4√x - 4 and g(x) = x - 1. Find the area A of R.

R is the region bounded by the functions f(x) = 1 – 2 cos (x) and g(x) = 4 sin (x). Find the area of the region bounded by the functions on the interval [0, π/2]. Enter an exact answer. Provide your answer below: A = ________units^2.

R is the region bounded by the functions f(x) = -2x + 6 and g(x) = -x/2 - 3 over the interval [a, b] where a = 2 and b = 5. Represent the area A of R by writing an integral with respect to x. You do not need to simplify.

R is the region bounded by the functions f(x) = 14/(-1 + x) and g(x) = 2x - 14. Find the area of the region bounded by the functions on the interval [6, 8]. Enter an exact answer. Provide your answer below: A = ________units^2.

Let f(x, y) = √1 -x²+y². (a) Determine the domain of f. (b) Identify the level curves and cross sections of f as conic sections (no sketches required).

R is the region bounded by the functions f(x) = 1 – 4√x and g(x) = -(4x/3) - (5/3). Find the area A of R. Enter an exact answer.

R is the region bounded by the functions f(x) = 3 - 2 cos (x) and 8(x) = 2sin(x). Find the area of the region bounded by the functions on the interval [0, π/2]. Enter an exact answer. Provide your answer below: A = _______units^2.

R is the region bounded by the functions f(x) = (12/x) and g(x) = (x/12). Find the area of the region bounded by the functions on the interval [1, 12]. Enter an exact answer. A = _____units^2.

R is the region bounded by the functions f(x) = 1 - 2cos(x) and g(x) = 4sin(x). Find the area of the region bounded by the functions on the interval [0, π/2]. Enter an exact answer. Provide your answer below: A = ________units^2.

Let f(x)=1/x The graph of f is given below. Determine the following function value. Enter DNE if the function is undefined at x = 0. f(0) =______

Please Simplify as instructed. sec(t) – cos(t) = (A) _________________ ( using sec(t) = 1 / cos(t) ) (B) _________________( adding using common denominator ) (C) __________________( Using sin²(t) + cos²(t) = 1 )

What is the exact value of tan (5π/6)? a. √3 b. -√3 c. √3/3 d. -√3/3

Use the Integral Test to determine whether the infinite series is convergent. Σ∞ n=1 1/(n+1)⁴ Σ∞ n=1 1/n+3 Σ∞ n=1 n‾¹/³ Σ∞ n=5 1/√n-4 Σ∞ n=25 n²/(n³ + 9)⁵/² Σ∞ n=1 n/(n² + 1)³/⁵

What is the area of AABC such that b = 28 centimeters, c = 16 centimeters, and m∠A= 25? a. 203.013 centimeters² b. 189.333 centimeters² c. 94.666 centimeters² d. 29.647 centimeters²

After 20 years, how much money will be in a fund that starts with $2500, and grows at a nominal rate of 6% per year, assuming the money is compounded: (a) Annually? (b) Weekly? (c) Continuously?

The interval in which the function f given by, f(x) = x² - 4x + 6 is increasing is (A) (-∞,2) (B) (2,∞) (C) (0,∞) (D) None of these

Let R be the region in the first quadrant bounded below by the parabola y = x^2 and above by the line y = 2. Then the value of ∫∫ yx dA is: A) 2/3 B) 4/3 C) 6 D) None of these.

Decide from the graph whether or not the integral ∫ e^(-4x)dx might converge. If the integral might converge, find out whether or not it does, and if so, the limit to which it converges to the nearest thousandth. Work:

Find the average value fave of the function f on the given interval. f(x)=√x, [0, 16]

Find the average value have of the function h on the given interval. In(u) U have 11 h(u): = [1, 7]

Analyze and then sketch the function Ax/x²+Bx+E (a) Determine the asymptotes. [A, 2] (b) Determine the end behaviour and the intercepts? [K, 2] (c) Find the critical points and the points of inflection. [A, 3] (d) Determine the interval of increasing/decreasing and the concavity. (e) Sketch the graph. [K, 2]

Evaluate each integral using a substitution. Clearly show the full substitution and be sure your limits always match the variable of integration. (a) ∫(1+tant)³ sec² t dt [0,π/4] (b) ∫e^ x/1+e^ 2x [0,1] (c)∫ sin(In x)√1+(In x)²/(x) dx [e,1/e]

The angle is labeled below. Find cos(beta). Give your answer in exact decimal format.

A curve passes through the point (0, 9) and has the property that the slope of the curve at every point P is twice the y-coordinate of P. What is the equation of the curve? (Use x as the independent variable.) y(x) = ________

Evaluate the definite integral. ∫√3/3 1/1+9x² dx

The region in the first quadrant between the curve y =. 2x² and the curve y -√x +5, is rotated about the x-axis. Find the integral for the volume of the resulting solid. You do not need to compute the volume. You may use your calculator to find the intersection point to the nearest hundredth.

Estimate ∫(sinx + cosx + 5)dx to 3 decimal places by using a left-hand Riemann sum with four subdivisions. How far from the exact value is your estimate?

Is the set of functions {1, sin x, sin 2x, sin 3x, ...) orthogonal on the interval [-π, π]? Justify your answer.

Evaluate the definite integral. ∫5/(1-25x^2)^(1/2)

A psychologist is studying the self image of smokers, as measured by the self-image (SI) score from a personality inventory. She would like to examine the mean SI score, μ, for the population of all smokers. Previously published studies have indicated that the mean SI score for the population of all smokers is 85 and that the standard deviation is 15, but the psychologist has good reason to believe that the value for the mean has decreased. She plans to perform a statistical test. She takes a random sample of SI scores for smokers and computes the sample mean to be 75. Based on this information, complete the parts below. (a) What are the null hypothesis Ho and the alternative hypothesis H, that should be used for the test? (b) Suppose that the psychologist decides to reject the null hypothesis. What sort of error might she be making? (Choose one) ▼ (c) Suppose the true mean SI score for all smokers is 71. Fill in the blanks to describe a Type II error. the hypothesis that u is (Choose one) A Type 11 error would be (Choose one) (Choose one) when, in fact, uis (Choose one) H <D 020 X x OSO 0=0 3 > 0

Graph the region bounded by the functions y = x² and y = x + 2, set up and evaluate the integral that will give the area.

Consider the solid region E bounded from above by the surface x²+y²+z²=4, bounded from the sides by the surface x²+y²=9y and bounded below by the xy- plane(z=0 ). Sketch the region and setup the integral in rectangular and cylindrical coordinates representing the volume of the solid.

Let h(x) = 8/x + 1. Use the limit definition of the derivative to differentiate h. h'(x) =______ Determine the slope of h at x = 8. h' (8) = ______

Suppose sin x = 32/68 and x is an angle in Quadrant 1. Then cos x = tan x =