# Integration using left, right, trapezoid, and midpoint rules

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Definite integrals

1. Show geometrically why int sqrt (2-x^2) dx = pi/4 + 1/2

2. Approximate int sqrt (2-x^2) dx for n = 5 using the left, right, trapezoid, and midpoint rules. Compute the error in each case using the answer to question 1 to compare the errors

https://brainmass.com/math/integrals/integration-using-left-right-trapezoid-midpoint-rules-67478

#### Solution Preview

Please see the attached file.

1.

Show geometrically why .

This is an equation of a circle with radius sqrt 2 as shown above.

The given integral is the area of the circle from x = 0 to x = 1 in the first quadrant . I have shaded this area in the first diagram.

Therefore the value of the integral = Area of the shaded region.

We will find this area using geometry. We need to find area ODAC.

Area ODAC =Area OBC - Area of OAB + Area of OAD(this is a ...

#### Solution Summary

Very detailed solutions are provided in a 3-page word document along with 2 diagrams and 3 charts.