While investigating ways to approximate definite integrals, it is insightful to compare results to integrals whose exact values we know. To that end, the following sequence of questions centers on $\int_0^3 x^2 \, dx$.

(a) Use the above applet to compute $L_3$, the left Riemann sum with three subintervals, of the function $f(x) = x^2$ on the window of $x$ values from $0$ to $3$.

$L_3=$

(b) Likewise, use the applet to compute $R_3$ and $M_3$, the right and middle Riemann sums with three subintervals, respectively.

$R_3=$

$M_3=$

(c) Use the Fundamental Theorem of Calculus to compute the exact value of $I = \int_0^3 x^2 \, dx$.

$I =$

(d) We define the error in an approximation of a definite integral to be the difference between the integral's exact value and the approximation's value. What is the error that results from using $L_3$? From $R_3$? From $M_3$?

$I-L_3 =$

$I-R_3 =$

$I-M_3 =$

(e) Let us develop a new approach to estimating the value of a definite integral known as the Trapezoid Rule. The basic idea is to use trapezoids, rather than rectangles, to estimate the area under a curve. What is the formula for the area of a trapezoid with bases of length $b$ and $B$ and height $h$?

Area of Trapizod $=$

(f) Working by hand, estimate the area under $f(x) = x^2$ on $[0,3]$ using three subintervals and three corresponding trapezoids. Call this value $T_3$.

$T_3 =$

What is the error in this approximation?

$I-T_3 =$

How does it compare to the errors you calculated in (d)?