Question Type 1: Estimating area with equal width intervals using trapezoids Exercises

Given the table below, approximate the area under the curve from $x=0$ to $x=4$ using the trapezoidal rule.

$x: 0, 1, 2, 3, 4$

$f(x): 2, 4, 7, 11, 16$

Use the trapezoidal rule with 3 equal subintervals to approximate the integral from $x=1$ to $x=4$ given the data below.

x: 1, 2, 3, 4
f(x): 10, 20, 25, 22

Approximate $\displaystyle \int_{0}^{1}e^{x}\,dx$ using the trapezoidal rule with 4 equal subintervals given the table below.

$x: 0, 0.25, 0.5, 0.75, 1$
$f(x)=e^{x}: 1, 1.2840, 1.6487, 2.1170, 2.7183$

Approximate $\displaystyle \int_{1}^{5}\frac{1}{x}\,dx$ using the trapezoidal rule with 4 equal subintervals.

The values of $\sin x$ are given in the table below at intervals of $0.5$ between $0$ and $2$. Use the trapezoidal rule to approximate $\displaystyle \int_{0}^{2}\sin x\,dx$.

$x: 0, 0.5, 1.0, 1.5, 2.0$

$f(x)=sin x: 0, 0.4794, 0.8415, 0.9973, 0.9093$

Use the trapezoidal rule with 3 subintervals to approximate $\displaystyle \int_{0}^{9}\sqrt{x}\,dx$.

Use the trapezoidal rule with 4 subintervals to approximate $\displaystyle \int_{0}^{\pi}\sin x\,dx$.

Approximate the integral $\displaystyle \int_{0}^{4}\bigl(5e^{-x}+e^{x}\bigr)\,dx$ using the trapezoidal rule with 2 equal subintervals.

The speed of a car (in m/s) is recorded at 5-minute intervals as shown. Use the trapezoidal rule to estimate the distance traveled over the 20 minutes.

time (min): $0, 5, 10, 15, 20$
speed (m/s): $0, 4, 7, 9, 10$

Approximate the integral $\displaystyle \int_{0}^{4}\bigl(5e^{-x}+e^{x}\bigr)\,dx$ using the trapezoidal rule with 4 equal subintervals.

Approximate the integral $\displaystyle \int_{0}^{4}\bigl(5e^{-x}+e^{x}\bigr)\,dx$ using the trapezoidal rule with 8 equal subintervals.