Ignacio and Kirsten want to find out if the triangle formed by connecting (-1,-1). (-3,2), and (2, 1) is a right triangle Ignaci
o's plan: . Use the distance formula to find the lengths of the sides. • Then see if the side lengths satisfy the equation a2 + b2 = c2 to see if it is a right triangle Kirsten's plan: • Find the slopes of the three sides. • See if any two slopes are negative reciprocals of each other Which student's plan will work?
1 answer:
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Split up the integration interval into 4 subintervals:
The left and right endpoints of the -th subinterval, respectively, are
for , and the respective midpoints are
- Trapezoidal rule
We approximate the (signed) area under the curve over each subinterval by
so that
- Midpoint rule
We approximate the area for each subinterval by
so that
- Simpson's rule
We first interpolate the integrand over each subinterval by a quadratic polynomial , where
so that
It so happens that the integral of reduces nicely to the form you're probably more familiar with,
Then the integral is approximately
Compare these to the actual value of the integral, 3. I've included plots of the approximations below.
The answer is what is shown on the phone
