Application of Integrals

The area enclosed by the curves \(y=x^3\) and \(y=\sqrt{x}\) is:

If the area of the region bounded by the curves \(y=4-\frac{{x}^{2}}{4}\) and \(y=\frac{x-4}{2}\) is equal to \(\alpha\), then \(6\alpha\) equals

[JEE Main 2025, 7 Apr (Shift 1)]

Let \(S={(x,y)\in \mathrm{ℝ}\times \mathrm{ℝ}:x\geq 0,y\geq 0,{y}^{2}\leq 4x,\) \({y}^{2}\leq 12-2x\) and \(3y+\sqrt{8}x\leq 5\sqrt{8}}\). If the area of the region S is \(\alpha \sqrt{2}\), then \(\alpha\) is equal to

The area enclosed by the curves \(y=x^3\) and \(y=\sqrt{x}\) is:

\(\text { The area of region enclosed by the curves } y=e^x, y=\left|e^x-1\right| \text { and } y \text {-axis is (in sq. units) }\)

The area of the region \(\left\{(\mathrm{x},\mathrm{y}):{\mathrm{x}}^{2}+4\mathrm{x}+2\leq \mathrm{y}\leq |\mathrm{x}+2|\right\}\) is equal to

[JEE Main 2025, 24 Jan (Shift 1)]

\(\text { The area of region enclosed by the curves } y=e^x, y=\left|e^x-1\right| \text { and } y \text {-axis is (in sq. units) }\)

The area of the region bounded by the parabola \((y-2)^2=\) \((x-1)\), the tangent to it at the point whose ordinate is 3 and the \(x\)-axis is:

[JEE Main 2021, 27 Aug (Shift 2)]

The area of the region

\(\left\{(x,y):0\leq x\leq \frac{9}{4},0\leq y\leq 1,x\geq 3y,x+y\geq 2\right\}\) is

Area (in sq. units) of the region outside \( \frac{|\mathrm{x}|}{2}+\frac{|\mathrm{y}|}{3}=1 \) and inside the ellipse \( \frac{x^{2}}{4}+\frac{y^{2}}{9}=1 \) is :

[JEE Main 2020, 2 Sep (Shift 1)]

The area, enclosed by the curves \(y=\text{sin}x+\text{cos}x\) and \(y=\left|\text{cos}x−\text{sin}x\right|\) and the lines \(x=0,x=\frac{\pi }{2}\) , is :

[JEE Main 2021, 1 Sep (Shift 2)]

The area of the region \(A=\left\{(x,y):\left|\cos x−\sin x\right|\leq y\leq \sin x,0\leq x\leq \frac{\pi }{2}\right\}\) is

[JEE Main 2023, 29 Jan (Shift 2)]