In Exercises 1318, write a Riemann sum and then a defi- nite integral representing the

(a) Write a Riemann sum approximating the area of the region in Figure 8.13, using vertical strips as shown. (b) Evaluate the corresponding integral.

(a) Write a Riemann sum approximating the area of the region in Figure 8.14, using vertical strips as shown. (b) Evaluate the corresponding integral.

(a) Write a Riemann sum approximating the area of the region in Figure 8.15, using horizontal strips as shown. (b) Evaluate the corresponding integral

(a) Write a Riemann sum approximating the area of the region in Figure 8.16, using horizontal strips as shown. (b) Evaluate the corresponding integral.

In Exercises 512, write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly

In Exercises 512, write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly

In Exercises 512, write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly

In Exercises 512, write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly

In Exercises 512, write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly

In Exercises 512, write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly

In Exercises 512, write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly

In Exercises 512, write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly

In Exercises 1318, write a Riemann sum and then a defi- nite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, and pyramids.)

In Exercises 1318, write a Riemann sum and then a defi- nite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, and pyramids.)

In Exercises 1318, write a Riemann sum and then a defi- nite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, and pyramids.)

In Exercises 1318, write a Riemann sum and then a defi- nite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, and pyramids.)

In Exercises 1318, write a Riemann sum and then a defi- nite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, and pyramids.)

In Exercises 1318, write a Riemann sum and then a defi- nite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, and pyramids.)

The integrals in Problems 1922 represent the area of either a triangle or part of a circle, and the variable of integration measures a distance. In each case, say which shape is represented, and give the radius of the circle or the base and height of the triangle. Make a sketch to support your answer showing the variable and all other relevant quantities.

The integrals in Problems 1922 represent the area of either a triangle or part of a circle, and the variable of integration measures a distance. In each case, say which shape is represented, and give the radius of the circle or the base and height of the triangle. Make a sketch to support your answer showing the variable and all other relevant quantities.

The integrals in Problems 1922 represent the area of either a triangle or part of a circle, and the variable of integration measures a distance. In each case, say which shape is represented, and give the radius of the circle or the base and height of the triangle. Make a sketch to support your answer showing the variable and all other relevant quantities.

The integrals in Problems 1922 represent the area of either a triangle or part of a circle, and the variable of integration measures a distance. In each case, say which shape is represented, and give the radius of the circle or the base and height of the triangle. Make a sketch to support your answer showing the variable and all other relevant quantities.

The integral , 1 0 (x x2 ) dx represents the area of a region between two curves in the plane. Make a sketch of this region.

In Problems 2427, construct and evaluate definite integral(s) representing the area of the region described, using: (a) Vertical slices (b) Horizontal slices

In Problems 2427, construct and evaluate definite integral(s) representing the area of the region described, using: (a) Vertical slices (b) Horizontal slices

In Problems 2427, construct and evaluate definite integral(s) representing the area of the region described, using: (a) Vertical slices (b) Horizontal slices

In Problems 2427, construct and evaluate definite integral(s) representing the area of the region described, using: (a) Vertical slices (b) Horizontal slices

The integrals in Problems 2831 represent the volume of either a hemisphere or a cone, and the variable of integration measures a length. In each case, say which shape is represented, and give the radius of the hemisphere or the radius and height of the cone. Make a sketch to support your answer showing the variable and all other relevant quantities.

The integrals in Problems 2831 represent the volume of either a hemisphere or a cone, and the variable of integration measures a length. In each case, say which shape is represented, and give the radius of the hemisphere or the radius and height of the cone. Make a sketch to support your answer showing the variable and all other relevant quantities.

The integrals in Problems 2831 represent the volume of either a hemisphere or a cone, and the variable of integration measures a length. In each case, say which shape is represented, and give the radius of the hemisphere or the radius and height of the cone. Make a sketch to support your answer showing the variable and all other relevant quantities.

The integrals in Problems 2831 represent the volume of either a hemisphere or a cone, and the variable of integration measures a length. In each case, say which shape is represented, and give the radius of the hemisphere or the radius and height of the cone. Make a sketch to support your answer showing the variable and all other relevant quantities.

Find the volume of a sphere of radius r by slicing.

Set up and evaluate an integral to find the volume of a cone of height 12 m and base radius 3 m.

Find, by slicing, a formula for the volume of a cone of height h and base radius r.

Figure 8.17 shows a solid with both rectangular and triangular cross sections. (a) Slice the solid parallel to the triangular faces. Sketch one slice and calculate its volume in terms of x, the distance of the slice from one end. Then write and evaluate an integral giving the volume of the solid. (b) Repeat part (a) for horizontal slices. Instead of x, use h, the distance of a slice from the top.

A rectangular lake is 150 km long and 3 km wide. The vertical cross-section through the lake in Figure 8.18 shows that the lake is 0.2 km deep at the center. (These are the approximate dimensions of Lake Mead, the largest reservoir in the US, which provides water to California, Nevada, and Arizona.) Set up and evaluate a definite integral giving the total volume of water in the lake.

A dam has a rectangular base 1400 meters long and 160 meters wide. Its cross-section is shown in Figure 8.19. (The Grand Coulee Dam in Washington state is roughly this size.) By slicing horizontally, set up and evaluate a definite integral giving the volume of material used to build this dam.

To find the area between the line y = 2x, the y-axis, and the line y = 8 using horizontal slices, evaluate the integral - 8 0 2y dy

The volume of the sphere of radius 10 centered at the origin is given by the integral - 10 10 102 x2 dx.

A region in the plane where it is easier to compute the area using horizontal slices than it is with vertical slices. Sketch the region.

A triangular region in the plane for which both horizontal and vertical slices work just as easily

In Problems 4245, are the statements true or false? Give an explanation for your answer

In Problems 4245, are the statements true or false? Give an explanation for your answer

In Problems 4245, are the statements true or false? Give an explanation for your answer

In Problems 4245, are the statements true or false? Give an explanation for your answer