Applications of Integrals — Class 12 Mathematics PYQs

The area of the shaded region bounded by the curves $y^2=x$, $x=4$ and the $x$-axis is given by

(A) ${\int }_0^{4}xdx$
(B) ${\int }_0^2{y}^{2}dy$
(C) $2{\int }_0^4\sqrt{x}dx$
(D) ${\int }_0^4\sqrt{x}dx$

The area of the shaded region (figure) represented by the curves $y=x^2$, $0\le x\le 2$ and $y$-axis is given by

(A) ${\int }_0^2{x}^{2}dx$
(B) ${\int }_0^2\sqrt{y}dy$
(C) ${\int }_0^4{x}^{2}dx$
(D) ${\int }_0^4\sqrt{y}dy$

Sketch the graph of $y=|x+3|$ and find the area of the region enclosed by the curve, $x$-axis, between $x=-6$ and $x=0$, using integration.

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

If $A_1$ denotes the area of region bounded by $y^2=4x$, $x=1$ and $x$-axis in the first quadrant and $A_2$ denotes the area of region bounded by $y^2=4x$, $x=4$, find $A_1:A_2$.

Using integration, find the area of the ellipse $\frac{x^2}{16}+\frac{y^2}{4}=1$, included between the lines $x=-2$ and $x=2$.

Find the area of the region bounded by the curves $x^2=y$, $y=x+2$ and $x$-axis, using integration.

Find the area of the region $\{(x,y):x^2+y^2\le 1\le x+y\}$, using integration.

Find the area of the region bounded by the lines $y=4x+5$, $x+y=5$ and $4y=x+5$, using integration.

Using Integration, find the area of triangle whose vertices are $(-1,1)$, $(0,5)$ and $(3,2)$.