Area between curves Quiz

1. Find the area enclosed by $y=x$ and $y=x^2$.

• A. $\frac13$
• B. $\frac12$
• C. $\frac16$
• D. $-\frac16$

2. Which integral gives the area enclosed by $y=4-x^2$ and $y=x+2$?

• A. $\int_{-2}^{1}(2-x-x^2)\,dx$
• B. $\int_{-2}^{1}(x^2+x-2)\,dx$
• C. $\int_{-1}^{2}(2-x-x^2)\,dx$
• D. $\int_{-2}^{1}(4-x^2)\,dx$

3. The graphs of $f$ and $g$ meet at $x=1$ and $x=3$, with $f\ge g$ between. Given $\int_1^3 f\,dx=14$ and $\int_1^3 g\,dx=9$, find the area between them.

• A. $23$
• B. $-5$
• C. $\frac{14}{9}$
• D. $5$

4. Evaluating $\int_{-1}^{1}(x-x^3)\,dx$ gives $0$. Why is this not the area between $y=x$ and $y=x^3$?

• A. The curves are not continuous.
• B. The line is above the curve on one side of $x=0$ and below it on the other, so the signed areas cancel.
• C. The integrand should be $x^3-x$.
• D. The limits should be $0$ and $1$ only.

5. Find the area enclosed by $y=6x-x^2$ and $y=x^2$.

• A. $18$
• B. $-9$
• C. $9$
• D. $\frac{27}{2}$