Application of Integrals Set-4

Q1. (Find the area of the region enclosed between the parabola $y = x^2 - 4x + 3$ and the line $y = 1$.)

Q2. (Find the area of the region enclosed between the curves $y = x^3 - x$ and $y = x$.)

Q3. (Find the area of the region enclosed by the curves $y = x^2$ and $x = y^2$.)

Q4. (Assertion (A): The area of the region bounded by the curve $y = x^{2}\sin\!\big(\tfrac{1}{x}\big)$ for $0<x\le 1$ and the $x$-axis is zero.
Reason (R): $\,\big|x^{2}\sin\!\big(\tfrac{1}{x}\big)\big|\le x^{2}$ and $\displaystyle\int_{0}^{1} x^{2}\,dx$ converges, so the region has finite area.)

Q5. (Compute the area of the region enclosed between the curves $y=\sin x$ and $y=\sin 2x$ on the interval $0\le x\le \pi$.)

Q6. Find the area enclosed between $y=x^2$ and $y=2x+3$.

Q7. The region bounded by $y=x^2$ and $y=4$ is rotated about the vertical line $x=-1$. The volume of the solid generated is

Q8. Let $R$ be the region in the first quadrant bounded by $y=x^2$ and $x=y^2$. Find the $x$-coordinate of the centroid of $R$.

Q9. The region bounded by $y=x$ and $y=x^2$ for $0\le x\le 1$ is revolved about the line $y=\dfrac12$. The volume of the resulting solid equals

Q10. Let $R$ be the region bounded by the parabola $y^2=4x$ and the line $y=x$ (with $y\ge0$). The $x$-coordinate of the centroid of $R$ is

Q11. Find the area of the region enclosed by the curves $y=2x$ and $y=x^2$.

Q12. The plane region bounded by $y=x$ and $y=x^2$ (for $0\le x\le1$) is considered. The $x$-coordinate of its centroid is

Q13. The region bounded by $y=x$ and $y=x^2$ for $0\le x\le1$ is revolved about the line $y=-1$. The volume of the solid generated is

Q14. Let $A$ be the area enclosed between $y=x$ and $y=x^2$. For which real values of $k$ does the area enclosed between $y=kx$ and $y=x^2$ equal $\dfrac{A}{2}$?

Q15. The region bounded by $y=x$ and $y=x^3$ for $0\le x\le1$ is revolved about the vertical line $x=2$. The volume of the resulting solid is