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

Let

By summing up the areas of trapezoids shown below,

we have

I hope that this was helpful.

By signing up, you agree to our Terms of Service and Privacy Policy

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.

By signing up, you agree to our Terms of Service and Privacy Policy

- How do you use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral #(1+cos(x))^(1/3)# from #[pi/2,0]?
- How do you Use a Riemann sum to approximate the area under the graph of the function #y=f(x)# on the interval #[a,b]#?
- How do you find the area using the trapezoidal approximation method, given #(5t + 6) dt #, on the interval [3, 6] with n=4?
- How do you Use Simpson's rule with #n=10# to approximate the integral #int_0^2sqrt(x)*e^(-x)dx#?
- Suppose f(x)= cos (x). How do you compute the Riemann sum for f(x) on the interval [0, (3pi/2)] obtained by partitioning into 6 equal subintervals and using the right hand end points as sample points?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7