The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. L1(2 -) - (*) dz
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- Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval. (Round your answers to four decimal places.) g(x) = 5x2 + 5, [1, 3], 8 rectangles _________________< area<____________________ this is a calculus problem. Can you write out each step neatly as possible and draw the graphs alsoPRACTICE ANOTHER Find the area (in square units) of the region under the graph of the function f on the interval [−1, 5]. f(x) = 2x + 4Sketch the region enclosed by the graphs of the functions f(x) = x^3and g(x) = 3x. Find thearea of this region. Justify your answer.
- Using rectangles each of whose height is given by the value ofthe function at the midpoint of the rectangle’s base (the midpoint rule),estimate the area under the graphs of the following functions, usingfirst two and then four rectangles. ƒ(x) = x3 between x = 0 and x = 1.g(x) = 2x2 − x − 1, [3, 5], 4 rectangles Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval.A farmer has 1,460feet of fencing available to enclose a rectangular area bordering a river. No fencing is required along the river. Let x represent the length of the side of the rectangular enclosure that is perpendicular to the river. function A(x), describes the total area of the rectangular enclosure as a function of x, where x is the length of the rectangular enclosure that is perpendicular to the river. A(x) = 1460x - 2x^2 The length of the side of the rectangle perpendicular to the river is? The length of the side of the rectangle parallel to the river is?
- A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of the function shown below and the x-axis (0 ≤ x ≤ 4) about the x-axis, where x and y are measured in meters. Use a graphing utility to graph the function. Find the volume of the tank. y=1/64x2 square root of 4-x2) Sketch the graphs of the functions f and g. f(x) = x2 + 2, g(x) = 1/3 x3 Find the area of the region enclosed by these graphs and the vertical lines x = −3 and x = 2. = ? square unitsShade the region under the graph of f(x)= 2/e^x bounded by x= -1 and x= 2 . Find the area of the shaded region.
- a) Estimate the area under the graph of f(x) = 7 + 4x2 from x = −1 to x = 2 using three rectangles and right endpoints. R3 = R6 = Sketch the curve and the approximating rectangles for R3 Sketch the curve and the approximating rectangles for R6. (b) Repeat part (a) using left endpoints. L3 = L6 = Sketch the curve and the approximating rectangles for L3. Sketch the curve and the approximating rectangles for L6. (c) Repeat part (a) using midpoints. M3 = M6 = Sketch the curve and the approximating rectangles for M3. Sketch the curve and the approximating rectangles for M6. (d) From your sketches in parts (a)-(c), which appears to be the best estimate?3) Sketch the graphs of the functions f and g. f(x) = square root x, g(x) = 1/2x -1 Find the area of the region enclosed by these graphs and the vertical lines x = 1 and x = 4. = ? square unitsA box with a square base and open top must have a volume of 143748 cm^3. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of x.]Simplify your formula as much as possible.A(x)=? Now, calculate when the function A(x) has a minimum.The length of the side of the square bottom is ? The minimum amount of material needed is?