Volumes and areas: harder questions Problem set A

In each part a curve is rotated 360° about an axis and the volume is given. Find the unknown limit.

1. (a) The part of the curve \(y=\frac{1}{x}\) between \(x=2\) and \(x=a\) (where \(a>2\)) is rotated about the x-axis. The volume of the solid is \(\frac{\pi}{3}\). Find the value of \(a\).
2. (b) The part of the parabola \(y=x^2\) between \(x=0\) and \(x=b\) (where \(b>0\)) is rotated about the y-axis. The volume of the solid is \(18\pi\). Find the value of \(b\).

The diagram shows the curve \(y=\sqrt{x+4}\), which meets the x-axis at the point A. The shaded region is bounded by the curve, the x-axis and the line \(x=6\).

1. (a) Write down the x-coordinate of A.
2. (b) The shaded region is rotated 360° about the x-axis. Find the volume of the solid.

The diagram shows the region bounded by the y-axis and the curve with equation \(x=9-y^2\).

1. (a) Find the area of the region.
2. (b) Find the volume of the solid formed when the region is rotated 360° about the y-axis.

Each solid is formed by rotating a region 360° about the y-axis.

1. (a) The region under the curve \(y=\frac1x\) between \(x=2\) and \(x=5\) (above the x-axis). Find the exact volume.
2. (b) The region bounded by the curve \(y=\sqrt{x-3}\), the line \(x=12\) and the x-axis. Find the volume to the nearest integer.

The region between the curve \(y=\sqrt{2x}\), the x-axis and the line \(x=8\) is rotated 360° about an axis.

1. (a) Sketch the curve for \(0\le x\le8\). Mark the end point.
2. (b) Find the volume when the region is rotated about the x-axis.
3. (c) Find the volume when the same region is rotated about the y-axis.

Find the exact volume of each solid. Each region is rotated 360° about the x-axis.

1. (a) The part of the curve \(y=\sin 3x\) between \(x=0\) and \(x=\frac{\pi}{3}\).
2. (b) The region bounded by the x-axis and the curves \(y=\sin 2x\) and \(y=\cos 2x\), for \(0\le x\le\frac{\pi}{4}\) (shown).

A lampshade is moulded in glass. All lengths are in cm. The curves \(y=x^2\) and \(y=\sqrt{8x}\) are shown.

1. (a) Find the coordinates of the points where the curves meet.
2. (b) The region between the curves is rotated 360° about the x-axis. Find the volume of the solid, in cm³.

The parabola \(y=2x^2\) and the line \(y=6x\) enclose a region, shown shaded.

1. (a) Find the points where the line meets the parabola.
2. (b) The region is rotated 360° about the y-axis. Find the volume of the solid.

The diagram shows the curve \(y=(3-x)e^{x/2}\) for \(x\ge0\). The curve meets the y-axis at A and the x-axis at B. The region R is bounded by the curve and the coordinate axes.

1. (a) Find the coordinates of A and B.
2. (b) Use technology to find the area of R.
3. (c) Use technology to find the volume of the solid formed when R is rotated 360° about the x-axis.

The cross-section of a ceramic cup is shaded in the diagram. The outer surface is \(y=0.25x^2\) and the inner surface is \(y=2+0.2x^2\), for \(x\ge0\). All lengths are in cm. The two surfaces meet at the rim.

1. (a) Find the height of the cup.
2. (b) The cup is rotated 360° about the y-axis. Find the volume of liquid it can hold.
3. (c) Find the volume of material used to make the cup.

The curve \(y=\sqrt{x^3+27}\) meets the coordinate axes at \(A(-3,0)\) and \(B(0,3 \sqrt{3})\). The region R is bounded by the curve, the x-axis and the y-axis.

1. (a) Show that \(x^2=\sqrt[3]{(y^2-27)^2}\).
2. (b) Find the area of R.
3. (c) Find the volume of the solid formed when R is rotated 360° about the x-axis.
4. (d) Find the volume of the solid formed when R is rotated 360° about the y-axis.

The diagram shows the curve \(y=3\ln x\). For \(a>1\), the region R is bounded by the curve, the coordinate axes and the line \(y=3\ln a\).

1. (a) Find, in terms of \(a\), the area of R.
2. (b) Hence show that \(\int_1^a 3\ln x\,dx=3a\ln a-3a+3\).

A cone has base radius \(r\) and height \(h\). Place its vertex at the origin with its axis along the x-axis.

1. (a) Write down the equation of the line through the origin and the point \((h,r)\).
2. (b) The part of this line between \(x=0\) and \(x=h\) is rotated 360° about the x-axis. Show that the volume of the cone formed is \(\frac13\pi r^2h\).
3. (c) Use the formula for a cone with radius 7 cm and height 15 cm.

A spherical cap is the part of a sphere on one side of a plane. Take a circle with centre the origin and radius r.

1. (a) Write down the equation of the circle.
2. (b) The region bounded by the upper half of the circle, the x-axis and the line \(x=a\) (where \(0<a<r\)) is rotated 360° about the x-axis. Show that the volume of the cap formed is \(\frac{\pi}{3}\left(2r^3-3r^2a+a^3\right)\).
3. (c) Find the volume of the cap when \(r=10\) cm and \(a=6\) cm.

The diagram shows the graph of \(y=x^4\) for \(x\ge0\), and two regions. Region A is under the curve between \(x=p\) and \(x=q\), where \(0<p<q\). Region B is bounded by the curve, the y-axis and the lines \(y=p^4\) and \(y=q^4\).

1. (a) Show that the ratio \(\text{area of A}:\text{area of B}\) does not depend on \(p\) or \(q\), and find the ratio.

The graph of \(y=\ln x\) is translated 1 unit to the left.

1. (a) Write down the equation of the translated curve.
2. (b) The region R is bounded by the translated curve, the y-axis and the line \(y=2\). Find the exact volume of the solid formed when R is rotated 360° about the y-axis.

Consider the curve \(y=2+\sin x\) for \(0\le x\le2\pi\).

1. (a) Sketch the curve.
2. (b) The curve is rotated 360° about the x-axis. Show that the volume of the solid is \(\frac{\pi}{2}\int_0^{2\pi}\left(9+8\sin x-\cos 2x\right)dx\).
3. (c) Find the exact value of the volume.