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

The following table shows the values of a function $f(x)$ for $0 \leq x \leq 4$.

$\begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & 2 & 4 & 7 & 11 & 16 \\ \hline \end{array}$

Approximate the area under the curve $y = f(x)$ from $x = 0$ to $x = 4$ using the trapezoidal rule.

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

Approximate $\displaystyle \int_{0}^{1} \text{e}^{x} \, \text{d}x$ using the trapezoidal rule with 4 equal subintervals, given the following data:

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

The values of $\sin x$ are given in the table below at intervals of $0.5$ for $0 \le x \le 2$.

Use the trapezoidal rule to find an approximate value for $\displaystyle \int_{0}^{2} \sin x \, \mathrm{d}x$.

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

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

Use the trapezoidal rule with 3 equal subintervals to approximate the integral $\int_{1}^{4} f(x) \, dx$ using the data provided in the table below:

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

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

The speed of a car $v$, in $\text{m}\,\text{s}^{-1}$, is recorded at $5$-minute intervals for $20$ minutes. The data is shown in the following table.

Use the trapezoidal rule to estimate the total distance traveled by the car during the $20$-minute period. Give your answer in metres ($\text{m}$).