In general, the midpoint approximation Mn forf (x)dx with n subdivisions is given by
with an error bound of
where K is any number such thatf′′(x)
≤K for all x∈
a, b
.
To approximate
x sin x dx using the midpoint method
Calculus + Approximate Integral + Midpoint
x sin x dx Approximate integral (midpoint rule) is
π
i +
πsin
i +
π
Evaluate Numerically
π
i +
πsin
i +
π = 3.1545
Note You can also find approximate integrals by specifying only the expression x sin x (without theor the dx), in which case you enter the lower and upper ends of the range in the dialog box.
To get the following output you enter 0 as Lower End of Range and 3.14159 as Upper End of Range in the dialog box.
Calculus + Approximate integral + Midpoint
x sin x Approximate integral (midpoint rule) is
.31416.31416i + .15708
sin
.31416i + .15708
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Evaluate
.31416.31416i + .15708
sin
.31416i + .15708
= 3.1545