Question: $16$. Consider $f(x)=x^2+1$.
$(a)$ Estimate the area under the graph of $f(x)$ on the interval $[0,6]$ using $3$ rectangles of equal width and right endpoints, as in the diagram below. Is your estimate an overestimate or an underestimate?
Graph shape: an increasing parabola $f(x)=x^2+1$ on the interval $[0,6]$, with three right-endpoint rectangles of equal width $2$; rectangle heights are based at $x=2$, $4$, and $6$, and the shaded region is shown under the curve. position_hint = center
1. **Problem statement:** Estimate the area under $f(x)=x^2+1$ on $[0,6]$ using $3$ rectangles of equal width with **right endpoints**, and decide whether the estimate is an overestimate or an underestimate.
2. **Use the right Riemann sum formula:**
$$R_n=\sum_{i=1}^{n} f(x_i)\,\Delta x$$
where $\Delta x=\frac{b-a}{n}$.
3. **Find the rectangle width:**
$$\Delta x=\frac{6-0}{3}=2$$
So each rectangle has width $2$.
4. **List the right endpoints:**
The subintervals are $[0,2]$, $[2,4]$, and $[4,6]$.
The right endpoints are $x=2$, $x=4$, and $x=6$.
5. **Evaluate the function at each right endpoint:**
$$f(2)=2^2+1=4+1=5$$
$$f(4)=4^2+1=16+1=17$$
$$f(6)=6^2+1=36+1=37$$
6. **Write the Riemann sum:**
$$R_3=2\big(f(2)+f(4)+f(6)\big)$$
$$R_3=2(5+17+37)$$
$$R_3=2(59)$$
$$R_3=118$$
7. **Decide overestimate or underestimate:**
Since $f(x)=x^2+1$ is **increasing** on $[0,6]$, right-endpoint rectangles lie **above** the curve on each subinterval.
So this is an **overestimate**.
8. **Final answer:** The estimated area is $118$, and it is an **overestimate**.