20.66
State the trapezoidal rule for approximation of integrals.
Let /(x)aO be integrable on [a, b]. Divide [a, b] into n equal parts of length A* = (fe — a)/n, by
means of points AC,,x 2 , ...,*„_,. Then
20.67
Use the trapezoidal rule with n = 10 to approximate
THE DEFINITE INTEGRAL AND THE FUNDAMENTAL THEOREM OF CALCULUS
159
[by Problem 20.65]
The actual value is
by the fundamental theorem of calculus.
20.68
State Simpson's rule for approximation of integrals.
Let f(x) be integrable and nonnegative on [a, b]. Divide [a, b\ into n=2k equal subintervals of length
Then
20.69
Apply Simpson's rule with n = 4 to approximate
We obtain, with
which
is close to the exact value
20.70
Use the trapezoidal rule with n = 10 to approximate
[by Problem 20.66]
which is close to the exact value
20.71
Use geometric reasoning to calculate
The graph of
is the upper half of the circle x~ + y
2 = a~ with center at the origin and
radius a.
is, therefore, the area of the semicircle, that is, -rra'12.
20.72
Find the area inside the ellipse
The area is twice the area above the x-axis and under the ellipse, which is given by
[by Problem 20.71]
Hence, the total area inside the ellipse is -nab.
20.73 Find
Hence,