Volumes of revolution

Figure 2 shows a sketch of part of the curve C with equation

The finite region R, shown shaded in Figure 2, is bounded by the line with equation x í WKH FXUYH C, the line with equation x = and the x-axis. The region R is rotated through 360° about the x-axis to form a solid of revolution with volume V. Use calculus to find the exact value of V, giving your answer in its simplest form. (Solutions based entirely on graphical or numerical methods are not acceptable.)

A container with a circular cross-section is shown in Figure 3. Initially the container is empty. At time t seconds after water begins to flow into the container, the height of water in the container is h cm. The height of water in the container satisfies the differential equation

where k is a constant. When h = 16, the height of water in the container is increasing at a rate of 0.6 cm s

Find the value of k.

Find the time taken to fill the container with water from empty to a height of 30 cm.

*LYHQ WKDW WKH ZDWHU IORZV LQWR WKH FRQWDLQHU DW D FRQVWDQW UDWH RI ³ s

find the volume of water in the container when h = 30 Give your answer in cm³ to 3 significant figures.

The finite region S, shown shaded in Figure 2, is bounded by the y-axis, the x-axis, the line with equation x = ln4 and the curve with equation

The region S is rotated through 2 radians about the x-axis. Use integration to find the exact value of the volume of the solid generated. Give your answer in its simplest form. [ Solutions based entirely on graphical or numerical methods are not acceptable. ]

Given that y 0, find

Use the substitution x = 4sin² to show that

Hence use integration to find

giving your answer in the form a + b, where a and b are exact constants.

Find

giving your answer in its simplest form.

Figure 3 shows a sketch of part of the curve C with equation

The curve C cuts the line y = 8 at the point P with coordinates (k, 8), where k is a constant.

Find the value of k.

The finite region S, shown shaded in Figure 3, is bounded by the curve C, the x-axis, the y-axis and the line y = 8. This region is rotated through 2 radians about the x-axis to form a solid of revolution.

Find the exact value of the volume of the solid generated.

Figure 3 shows a sketch of part of the curve C with equation

The point P lies on C and has coordinates (2, 9). The line l is a tangent to C at P. The line l cuts the x-axis at the point Q.

Find the exact value of the x coordinate of Q.

The finite region R, shown shaded in Figure 3, is bounded by the curve C, the x-axis, the y-axis and the line l. This region R is rotated through 360° about the x-axis.

Use integration to find the exact value of the volume of the solid generated. Give your answer in the form where p and q are exact constants. [You may assume the formula V = r₂h for the volume of a cone.]

The finite region R is bounded by the curve C, the x- axis, the line with equation x = 1 and the line with equation x = 4 This region is shown shaded in Figure 2 The region R is rotated through 360 ° about the x- axis.

Use calculus to find the exact volume of the solid of revolution generated, giving your answer in the form a + b ln c, where a, b and c are constants.

Figure 4 shows a sketch of part of the curve C with parametric equations

The point P lies on C and has coordinates (3, 2). The line l is the normal to C at P. The normal cuts the x-axis at the point Q.

Find the x coordinate of the point Q.

The finite region S, shown shaded in Figure 4, is bounded by the curve C, the x-axis, the y-axis and the line l. This shaded region is rotated 2 radians about the x-axis to form a solid of revolution.

Find the exact value of the volume of the solid of revolution, giving your answer in the form + q², where p and q are rational numbers to be determined. [You may use the formula V πr²h for the volume of a cone.]

Figure 1 shows the finite region R bounded by the x-axis, the y-axis, the line x and the curve with equation

Complete the table above giving the missing value of y to 6 decimal places.

Using the trapezium rule, with all of the values of y from the completed table, find an approximation for the area of R, giving your answer to 4 decimal places.

Region R is rotated through 2 radians about the x-axis.

Use calculus to find the exact volume of the solid formed.

Figure 2 showsa sketch of the curve C with parametric equations

Show that the cartesian equation of C may be written in the form

stating the values of a and b. (3)

shown shaded in Figure 3. This region is rotated through 27 radians about the x-axis to form a solid of revolution.

Use calculus to find the exact value of the volume of the solid of revolution. (5)

Figure 2 shows a sketch of part of the curve C with parametric equations

The curve crosses the y-axis at the point A and crosses the x-axis at the point B.

Show that A has coordinates (0, 3).

Find the x coordinate of the point B.

Find an equation of the normal to C at the point A.

The region R, as shown shaded in Figure 2, is bounded by the curve C, the line x = –1 and the x-axis.

Use integration to find the exact area of R.

Figure 3 shows a sketch of part of the curve with equation y = 1 – 2cos x, where x is measured in radians. The curve crosses the x-axis at the point A and at the point B.

Find, in terms of , the x coordinate of the point A and the x coordinate of the point B.

The finite region S enclosed by the curve and the x-axis is shown shaded in Figure 3. The region S is rotated through 2 radians about the x-axis.

Find, by integration, the exact value of the volume of the solid generated.

Figure 3 shows part of the curve C with parametric equations

The point P lies on C and has coordinates ˲ 3, 3.

Find the value of at the point P.

The line l is a normal to C at P. The normal cuts the x-axis at the point Q.

Show that Q has coordinates k ˲ 3, 0, giving the value of the constant k.

The finite shaded region S shown in Figure 3 is bounded by the curve C, the line x ˲3 and the x-axis. This shaded region is rotated through 2 radians about the x-axis to form a solid of revolution.

Find the volume of the solid of revolution, giving your answer in the form p ¥ 3 + q ², where p and q are constants.

The curve C has parametric equations

Find

a cartesian equation of C.

The finite area R, shown in Figure 1, is bounded by C, the x-axis, the line x ln 2 and the line x ln 4. The area R is rotated through360 about the x-axis.

Use calculus to find the exact volume of the solid generated.

Using the substitution x = 2 cos u, or otherwise, find the exact value of

The shaded region S, shown in Figure 3, is bounded by the curve, the x-axis and the lines with equations x = 1 and x = √2. The shaded region S is rotated through 2π radians about the x-axis to form a solid of revolution.

Using your answer to part (a), find the exact volume of the solid of revolution formed.

Figure 1 shows part of the curve y The region R is bounded by the curve, the x-axis, and the lines x = 0 and x = 2, as shown shaded in Figure 1.

Use integration to find the area of R.

The region R is rotated 360° about the x-axis.

Use integration to find the exact value of the volume of the solid formed.

curve, the x-axis and the lines x = a and x = b is shown shaded in Figure 2. This region is rotated through 360° about the x- axis to generate a solid of revolution. Find the volume of the solid generated. Express your answer as a single simplified fraction, in terms of a and b.