Consider the following limit. limx→5 (4x + 5) Find the limit L. L = (a) Find the largest δ > 0 such that |f(x) − L| < 0.01 whenever 0 < |x − c| < δ. (Round your answer to five decimal places.) δ = (b) Find the largest δ > 0 such that |f(x) − L| < 0.005 whenever 0 < |x − c| < δ. (Round your answer to five decimal places.) δ =
Find the surface area of the pyramid. The surface area of the pyramid is m2. (Do not round until the final answer. Then round to the nearest whole number as needed.)
Find the area bounded by the given curve using the vertical strip (y dx) and the horizontal (x dy) strip. y = x2, y = 2x + 3
Let f(x) = 4x − x2, 1 ≤ x ≤ 2, and let S be the region that lies under the graph of f and above the x-axis. Estimate the area of the region S using five approximating rectangles, and right endpoints. What is the estimated area? Do not round the answer. Type the value of the estimated area. Answer: Check
Write a definite integral that represents the area of the region. (Do not evaluate the integral.) y1 = 7(x3 − x) y2 = 0 ∫−1 0 ( )dx
Find the area between y = −x4 + 4x2 + 2, y = x − 2, and −1.8 ≤ x ≤ 1.8. Round your limits of integration and answer to 2 decimal places. The area between the curves is square units.
Part A Locate the centroid y¯ of the area. Express your answer in terms of some or all of the variables a, n, and h. y¯ = Submit Request Answer
Two airplanes are flying in the air at the same height. Airplane A is flying east at 230 mi/h and airplane B is flying north at 270 mi/h. a. Take the derivative of the Pythagorean Theorem, a2 + b2 = c2 with respect to t. b. If they are both heading to the same airport, located 44 miles east of airplane A and 33 miles north of airplane B, at what rate is the distance between the airplanes changing? Answer exactly. The distance is shrinking at a rate of mi/h. Hint: The phrase "shrinking at a rate" implies the rate of change is negative so do not enter a negative number as your answer.
1.2 Determine the area of region A, bounded by the curve defined by y = x2 − 4 and the lines y = 0 and x = 4, as shown in the diagram below.
Find the area of the region enclosed by y = 1.75 x and x = 9 − y2. Use horizontal strips to find the area. First find the y coordinates of the two points where y = 1.75 x meets x = 9−y2. lower y = c = and upper y = d = Hint: g(y) is the right curve - left curve. Then area of the region = ∫c d g(y)dy where g(y) = Evaluate the definite integral to find Area =
Determine the area of the shaded region in the following figure by integrating with respect to y. The area of the shaded region is
Find the area between y = 1, y = −.75(x − 4.5)2 + 3, and x ≥ 0. If needed, round your limits of integration and answer to 2 decimal places. The area between the curves is square units.
Let the region R be the area enclosed by the function f(x) = x3 + 2, the x-axis and the vertical lines x = 0 and x = 2. Find the volume of the solid generated when the region R is revolved about the x-axis. You may use a calculator and round to the nearest thousandth.
For the following exercises, split the region between the two curves into two smaller regions, then determine the area by integrating over the x-axis. Note that you will have two integrals to solve. 3. y = x3 and y = x2 + x
Find the area between the following curves. x = −6, x = 1, y = ex, and y = 1−ex The area is (Type an exact answer in terms of e. )
Each of the regions A, B, and C bounded by the graph of f and the x-axis has area 3 . Find the value of ∫−4 2 [f(x) + 2x + 3]dx.
Find the area A of the region inside the circle r = 4 sin(θ + π/4) and above the line r = 2 sec(θ − π/4). (Use symbolic notation and fractions where needed. ) A =
The shaded area shown in (Figure 1) is bounded by x, y axes and the curve y2 = 3.24 − 0.5x m2, where x is in m. Suppose that a = 6.48 m and h = 1.8 m. Figure 1 of 1 Determine the moment of inertia for the shaded area about the y axis. Express your answer to three significant figures and include the appropriate units. Iy =
Set up the definite integral that would allow you to find the area of the region enclosed between y = 3x − 2 and y = −x2 + 4x + 4 which is sketched below. Note: You do not need to evaluate the definite integral.
Calculate the definite integral by referring to the figure with the indicated areas. ∫0 c f(x)dx Area A = 1.348 Area C = 5.475 Area B = 2.309 Area D = 1.662 ∫0 c f(x)dx =
Find the area of the region bounded above by the parabola y = x2 + 2 and below by the line y = x where −1 ≤ x ≤ 2.
Find the volume of the figure. Express the answer in terms of π and then round to the nearest whole number. The volume of the figure is exactly (Type an exact answer in terms of π Simplify your answer.)
Volume of a Torus A torus is formed by revolving the region bounded by the circle x2 + y2 = 1 about the line x = 2 (see figure). Find the volume of this "doughnut-shaped" solid. (Hint: The integral ∫−1 1 1−x2 dx represents the area of a semicircle.)
Find the volume of the shaded region. When appropriate, use the π key on your calculator and round your answer to the nearest hundredth. The volume of the region is (Round to the nearest hundredth as needed.)
Find the volume under the given surface z = f(x, y) and above the rectangle with the given boundaries. z = 2x2 + 7; 0 ≤ x ≤ 4, 0 ≤ y ≤ 3 What is the volume of the region? units 3 (Simplify your answer. Type an exact answer. )
Use the shell method to find the volume of the solid generated by revolving the regions bounded by the curves and lines about the y-axis. y = 98 − x2, y = x2, x = 0 for x ≥ 0 The volume of the solid is cubic units. (Type an exact answer in terms of π.)
Use Disks method to find the volume of the solid that results when the region enclosed by the curves is revolved about the x-axis.
Write six different iterated triple integrals for the volume of the tetrahedron cut from the first octant by the plane 6x + 3y + 2z = 6. Evaluate the first integral. Write an integral that represents the volume in the order dx dy dz.
Find the volume of the following solid beneath the paraboloid f(x, y) = 9 − x2 − 2y2 and f(x, y) = 9 − x2 − 2y2 above the region R = {(x, y): 1 ≤ x ≤ 2, 0 ≤ y ≤ 1}. The volume bounded by the paraboloid is (Simplify your answer. )
The volume, of a rectangular solid with length L, width W, and height H is given by the formula V = LWH. Use this formula to write a polynomial in standard form that models, or represents, the volume of the open box. The volume of the open box is (Simplify your answer. Type your answer in standard form.)
Find the volume of the solid that results when the region enclosed by x = −y3 + 3, x = 2, y = −1 and y = 0 is revolved around x = −1. Volume = units 2
The diagram below shows a part of the curve y = (1 + 4x)12 and a point A(2, 3) lying on the curve. The line AB intersects the x-axis at B(5, 0). Fill in the blank with the correct number. The volume of the solid of revolution obtained when the shaded region is rotated through 360∘ about the x-axis is π cubic units.
Find the volume of the solid generated by revolving the following region about the given axis. The region in the first quadrant bounded above by the curve y = x2, below by the x-axis, and on the right by the line x = 1, about the line x = −2. Use the washer method to set up the integral that gives the volume of the solid. V = ∫ dx
Find the volume of the figure. Express the answer in terms of π and then round to the nearest whole number. The volume of the figure is exactly (Type an exact answer in terms of π. Simplify your answer. ) The volume of the figure is approximately (Round to the nearest whole number as needed.)
Find the volume of the parallelepiped (box) determined by u, v, and w. u = 4i + 2j, v = 3i − 2j + 4k, w = i + k The volume of the parallelepiped is units cubed. (Simplify your answer. )
Referring to the figure above, find the volume generated by rotating the region R1 about the line OC. Volume =
Referring to the figure above, find the volume generated by rotating the region R1 about the line OC. Volume =
Find the volume of the solid generated by revolving the shaded region about the x-axis. The volume of the solid is cubic units. (Type an exact answer, using π as needed.)
Find the surface area of the pyramid. The surface area of the pyramid is in2 (Type a whole number.)
Find the area of the following surface using a parametric description of the surface. The plane z = 13−x−y above the square |x| ≤ 3, |y| ≤ 3 Set up the integral for the surface area using the parameterization u = x and v = y. ( )du dv (Type an exact answers, using π as needed. ) The area is (Type an exact answer, using radicals as needed.)
Use technology to find the surface area S of the surface generated when the plane curve defined by the equations x(t) = −t2 − 3t + 1, y(t) = t3 + 3t2 + t − 2, on the interval 2 ≤ t ≤ 4, is revolved around the x-axis. If necessary, round your answer to two decimal places. Provide your answer below: S ≈
Find the area of the surface generated when the given curve is revolved about the given axis. y = 4x + 6, for 0 ≤ x ≤ 3; about the x-axis The area of the surface is square units. (Type an exact answer, using π as needed.)
Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis. (Round your answer to three decimal places.) 2π∫4 y = 2x dx =
Use the Midpoint Rule to estimate the surface area of the pond shown in the figure, where x = 10. (Round your answer to the nearest integer.) ft2
Find the surface area of the cylinder. The surface area of the given cylinder is in2. (Simplify your answer. Type an exact answer in terms of π. Use integers or decimals for any numbers in the expression.)
The region bounded by x = 0, y = 2x and y = 3 − x2 is revolved around the x-axis it is easier to find the volume of resulting using shell method Select one: True False
Using both the disk/washer method and the shell method, find the volume of the solid obtained by revolving each region about the indicated axis. a) Region bounded by xy = 1, y = 0, x = 1, and x = 2 revolved about the line x = −1 b) Region bounded by y = x and y = x3 revolved about the line y = 2
Find the volume of the solid generated when the indicated region is revolved about the specified axis; slice, approximate, integrate. x-axis 16π 14π 8π 4π
Let the region R be the area enclosed by the function f(x) = ex and g(x) = 6x + 1. Find the volume of the solid generated when the region R is revolved about the x-axis. Yóu may use a calculator and round to the nearest thousandth
Write and evaluate the definite integral that represents the volume of the solid formed by revolving the region about the y-axis. y = x 2/3
Consider the region R bounded by the graphs of y = (x) > 0, x = a > 0, x = b > a, and y = 0. (a) Suppose f(x) = 4x2 − x4 with a = 1, b = 2. Calculate the volume of the solid formed by revolving R about the x-axis. (b) Do the same for the solid formed by revolving R about the y-axis