Question
Determine the area under the cutve $\text{y}=\sqrt{\text{x}^{2}-\text{x}^{2}}$ included between the lines $x = 0$ and $x = a$.
✓
Answer
We have,
$\text{y}=\sqrt{\text{a}^{2}-\text{x}^{2}}$
$\Rightarrow \text{y}^{2}={\text{a}^{2}-\text{x}^{2}}$
$\Rightarrow \text{x}^{2}+\text{y}^{2}=\text{a}^{2}$
Since in the given equation $x^2 + y^2 = a^2$
all the powers of both x and y are even, the curve is symmetrical about both the axis.
$\therefore$ Area encloed by the curve and above x-axis = area A'BA = 2 × area enclosed by ellipse and x-axis in first quadrant
(a, 0), (-a, 0) are the points of intersection of curve and x-axis
(0, a), (0, a) are the points of intersection of curve and x-axis
Slicling the area in the first quadrant into vertical stripes of height = |y| and width = dx
$\therefore$ Area of approximating rectangle = |y| dx
Appoximating rectangle can move between x = 0 and x = a
A = Area of enclosed curve in the first $=\int\limits_{0}^{a}|\text{y}|\text{dx} $
$\Rightarrow \text{A}=\int\limits_{0}^{a}\sqrt{\text{a}^{2}-\text{x}^{2}}\text{dx}$
$\Rightarrow \Bigg[\frac{1}{2}\text{x}\sqrt{\text{a}^{2}-\text{x}^{2}}+\frac{1}{2}\text{a}^{2}\sin^{-1}\frac{\text{x}}{\text{a}}\Bigg]^{\text{a}}_{0}$
$=\frac{1}{2}\text{a}^{2}\sin^{-1}1$
$=\frac{1}{2}\text{a}^{2}\frac{\pi}{2}$
$=\frac{\text{a}^{2}\pi}{4}\ \text{sq.}\ \text{units}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If $\text{y}=\log\big\{\sqrt{\text{x}-1}-\sqrt{\text{x}+1}\big\},$ show that $\frac{\text{dy}}{\text{dt}}=\frac{-1}{2\sqrt{\text{x}^2-1}}.$
→In a bulb factory, machines A, B and C manufacture $60 \%, 30 \%$ and $10 \%$ bulbs respectively. $1 \%, 2 \%$ and $3 \%$ of the bulbs produced respectively by $\mathrm{A}, \mathrm{B}$ and C are found to be defective. A bulb is picked up at random from the total production and found to be defective. Find the probability that this bulb was produced by the machine A.
→$\text{Let A = Q}\times\text{Q},$ where Q is the set of all rational numbers and $\ast$ be a binary operation defined on A by
$\text{(a, b)}\ast\text{(c, d) = (ac, b + ad)}$ for all $\text{(a, b) (c, d)}\in \text{A}.$
→$\text{(a, b)}\ast\text{(c, d) = (ac, b + ad)}$ for all $\text{(a, b) (c, d)}\in \text{A}.$
- The identity element in A.
- The invertible element of A.
Discuss the continuity of the f(x) at the indicated points f(x) = |x| + |x - 1| at x = 0, 1.
Maximize Z = 9x + 3y
Subject to
$2\text{x}+3\text{y}\leq13$
$3\text{x}+\text{y}\leq5$
$\text{x},\text{y}\geq0$
→Subject to
$2\text{x}+3\text{y}\leq13$
$3\text{x}+\text{y}\leq5$
$\text{x},\text{y}\geq0$
Discuss the applicability of Lagrange's mean value theorem for the function:
f(x) = |x| on [−1, 1]
f(x) = |x| on [−1, 1]
Differentiate $\tan^{-1}\Big(\frac{\cos\text{x}}{1+\sin\text{x}}\Big)$ with respect to $\sec^{-1}\text{x}$
→Evaluate the following intregals:
$\int\frac{1}{\text{x}\log\text{x}(2+\log\text{x})}\text{ dx}$
→$\int\frac{1}{\text{x}\log\text{x}(2+\log\text{x})}\text{ dx}$
Evaluate the following integrals as limit of sum:
$\int\limits^5_{3}(2-\text{x})\text{dx}$
→$\int\limits^5_{3}(2-\text{x})\text{dx}$
If $\text{x}=\cos\text{t}(3-2\cos^2\text{t}),\text{y}\sin\text{t}(3-2\sin^2\text{t})$ find the value of $\frac{\text{dy}}{\text{dx}}\text{ at t}=\frac{\pi}{4}$
→