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

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

Find the area between the curve \(y=x^{3}\), the y-axis and the lines \(y=1\) and \(y=64\).

Find the area between the curve \(y=e^{0.5x}\), the y-axis and the lines \(y=2\) and \(y=7\).

Find the area between the curve \(y=\frac{7}{x-3}\), the y-axis and the lines \(y=2\) and \(y=6\).

Find the area between the curve \(y=\tan x\), the y-axis and the lines \(y=1.5\) and \(y=5\). Use radians.

Find the area between the curve \(y=5\sin x\), the y-axis and the lines \(y=1\) and \(y=4\), for \(0\le x\le\frac{\pi}{2}\).

The side of a grass embankment beside a cycle path is curved, with lengths in metres. The region is bounded by the curve \(y=e^{0.5x}\), the y-axis and the lines \(y=1.5\) and \(y=8\). Find its area in m².

The glass of a lamp shade has a curved cross-section, with lengths in centimetres. Find the area between the curve \(y=\frac{9}{x-1}\), the y-axis and the lines \(y=2\) and \(y=6\), in cm².

The diagram shows the curve \(y=\ln x\). The shaded area is \(\int_{1}^{5}\ln x\,dx\).

1. (a) Find the area between the curve \(y=\ln x\), the y-axis and the lines \(y=0\) and \(y=\ln 5\).
2. (b) Use the rectangle with corners \((0,0)\) and \((5,\ln 5)\) to show that \(\int_{1}^{5}\ln x\,dx=5\ln 5-4\).

A banner is cut from a region of cloth. The area between the curve \(y=\sqrt{x}\), the y-axis and the lines \(y=2\) and \(y=k\), where \(k>2\), is 25. Find k.