Area under the curveMedium
Question
The area of the region $\{(x,y):y \leq \pi - |x|,y \leq \mid xsinx \mid ,y \geq 0\}$ is :
Options
A.$1 + \frac{\pi^{2}}{8}$
B.$2 + \frac{\pi^{2}}{4}$
C.$\frac{\pi^{2}}{8} - 1$
D.$4 + \frac{\pi^{2}}{2}$
Solution
Sol. [Key Concept: Sketch $y = \pi - |x|$ and $y = |xsin\ x|$; integrate the enclosed strips.]
Area ; $2\left( \int_{0}^{\pi/2}\mspace{2mu} xsinxdx + \int_{\pi/2}^{\pi}\mspace{2mu}(\pi - x)dx \right)$
$$\begin{matrix} & \ = 2( - xcosx + sinx)_{0}^{\pi/2} + 2\left( \left. \ \frac{(\pi - x)^{2}}{2} \right|_{\pi}^{\pi/2} \right) \\ & \ = 2 + \frac{\pi^{2}}{4} \end{matrix}$$
Create a free account to view solution
View Solution FreeMore Area under the curve Questions
The area bounded by circle x2 + y2 = 4 ;curve y = (where [ . ] is G.I.F.) and the x-axis ; is...Area bounded by the curve y = x sinx and x-axis between x = 0 and x = 2π is...The area (in sq. units) of the region described by {(x, y) : y2 ≤ 2x and y ≥4x − 1} is :...The area bounded by the curve y = (x − 1)(x − 2) (x − 3), x-axis and odinates x = 0,x = 3 is-...One vertex of the triangle of maximum area that can be inscribed in the curve |z - 2i| = 2 is 2 + 2i, remaining vertices...