Area between a curve and the y-axis Problem set A

Find the area between the curve \(y=\sqrt{x}\), the y-axis and the lines \(y=2\) and \(y=4\).

Find the area between the curve \(y=x^{3}\), the y-axis and the lines \(y=8\) and \(y=27\).

Find the area between the curve \(y=e^{5x}\), the y-axis and the lines \(y=3\) and \(y=8\).

Find the area between the curve \(y=\frac{3}{x-2}\), the y-axis and the lines \(y=1\) and \(y=4\).

Find the area between the curve \(y=\tan x\), the y-axis and the lines \(y=0.5\) and \(y=3\). Use radians.

Find the area between the curve \(y=2\sin x\), the y-axis and the lines \(y=0.5\) and \(y=1.5\), for \(0\le x\le\frac{\pi}{2}\).

A skate ramp has a curved side panel, with lengths in metres. The region is bounded by the curve \(y=e^{0.4x}\), the y-axis and the lines \(y=2\) and \(y=9\). Find its area in m².

The wall of a plastic funnel is shaped like a curve, with lengths in centimetres. Find the area between the curve \(y=\frac{6}{x-2}\), the y-axis and the lines \(y=3\) and \(y=8\), in cm².

The diagram shows the curve \(y=\ln x\). The shaded area is \(\int_{1}^{7}\ln x\,dx\).

1. (a) Find the area between the curve \(y=\ln x\), the y-axis and the lines \(y=0\) and \(y=\ln 7\).
2. (b) Use the rectangle with corners \((0,0)\) and \((7,\ln 7)\) to show that \(\int_{1}^{7}\ln x\,dx=7\ln 7-6\).

A sail is cut from a region of cloth. The area between the curve \(y=\sqrt{x}\), the y-axis and the lines \(y=1\) and \(y=k\), where \(k>1\), is 18. Find k.