Browse Questions

# Using integration,find the area of the region bounded by the line $2y=5x+7,x$-axis and the lines $x=2$ and $x=8.$

Toolbox:
• The area enclosed by the curve $y=f(x)$,the $x$-axis and the ordinates $x=a$ and $x=b$ is given by $\int_a^b ydx.$
• $\int x^n dx=\large\frac{x^{n+1}}{n+1}$$+c. Step 1: Given 2y=5x+7 and x=2 and x=8. The area of the required region is shown in the fig. The required area is A=\int_2^8\large\frac{5x+7}{2}$$dx$
$\qquad\qquad\qquad\qquad=\large\frac{1}{2}$$\int_2^8(5x+7)dx. Step 2: On integrating we get, \qquad\qquad\qquad\qquad=\large\frac{1}{2}$$\begin{bmatrix}\large\frac{5x^2}{2}\normalsize+7x\end{bmatrix}_2^8.$
On applying limits we get,
$\qquad\qquad\qquad\qquad=\large\frac{1}{2}$$[5\times \big(\large\frac{8^2}{2}\big)-$$5\big(\large\frac{2^2}{2}\big)$$+7(8)-7(2)] \qquad\qquad\qquad\qquad=\large\frac{1}{2}$$[5\times 32-10+56-14]$
$\qquad\qquad\qquad\qquad=\large\frac{1}{2}$$[192]$
$\qquad\qquad\qquad\qquad=96$sq.units.
Hence the required area is $96$sq.units.