Area under a curve given parametrically Quiz

1. $x=t^3$, $y=t^2$. Find the area under the curve for $0\le t\le1$.

• A. $\frac13$
• B. $\frac35$
• C. $\frac12$
• D. $\frac15$

2. $x=2t$, $y=t^2$. Find the area under the curve from $x=0$ to $x=4$.

• A. $\frac83$
• B. $4$
• C. $\frac{64}{3}$
• D. $\frac{16}{3}$

3. Which integral gives the area under $x=f(t)$, $y=g(t)$ from $t=\alpha$ to $t=\beta$?

• A. $\int_\alpha^\beta y\frac{dx}{dt}\,dt$
• B. $\int_\alpha^\beta y\,dt$
• C. $\int_\alpha^\beta x\frac{dy}{dt}\,dt$
• D. $\int_\alpha^\beta y^2\frac{dx}{dt}\,dt$

4. $x=t+1$, $y=e^t$. Find the area under the curve from $x=1$ to $x=3$.

• A. $e^3-e$
• B. $e^2$
• C. $e^2-1$
• D. $e^2+1$

5. $x=t^2$, $y=\frac1t$. Find the area under the curve for $1\le t\le3$.

• A. $2$
• B. $\ln3$
• C. $4$
• D. $6$