How do you Use the trapezoidal rule with #n=10# to approximate the integral #int_0^2sqrt(x)*e^(-x)dx#?

Question

Ezra Adams · Accepted Answer

Let #f(x)=sqrt{x}e^{-x}=sqrt{x}/e^x#.

#Delta x={b-a}/n={2-0}/10=0.2#

By summing up the areas of trapezoids shown below,

we have

#T_{10}=[f(0)+2f(0.2)+2f(0.4)+cdots+2f(1.8)+f(2)]{Delta x}/2approx 0.64#

I hope that this was helpful.

Adam Adkins · Answer

undefined To use the Trapezoidal Rule with $ n = 10 $ to approximate the integral $ \int_{0}^{2} \sqrt{x} \cdot e^{-x} \, dx $, you first need to divide the interval $[0, 2]$ into $ n = 10 $ subintervals of equal width. Then, you evaluate the function at the endpoints of these subintervals and use the formula for the Trapezoidal Rule to calculate the approximate area under the curve.

The formula for the Trapezoidal Rule is:

$$ \text{Approximate area} = \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] $$

Where:
- $ h $ is the width of each subinterval, given by $ \frac{b - a}{n} $ for an interval $[a, b]$.
- $ x_0 $ and $ x_n $ are the endpoints of the interval.
- $ x_i $ are the points within the interval, evenly spaced.

For this problem:
- $ a = 0 $ and $ b = 2 $.
- $ h = \frac{2 - 0}{10} = 0.2 $.

Now, calculate the function values at the endpoints and within the subintervals $ x_1, x_2, ..., x_9 $, then apply the Trapezoidal Rule formula to find the approximate area under the curve.