Question

Find an approximation of the area of the region R under the graph of the function f on the interval [0, 2]. Use n = 5 subintervals. Choose the representative points to be the midpoints of the subintervals.

f (x) = x^2 + 5

Answer 1

Interval:
`[0,2]`

Partition:
`\left[0,\frac25\right]\cup\left[\frac25,\frac45\right]\cup\left[\frac45,\frac65\right]\cup\left[\frac65,\frac85\right]\cup\left[\frac85,2\right]`

Midpoints:
`\left\{\frac15,\frac35,1,\frac75,\frac95\right\}`

Value of
`f(x)` at the midpoints:
`\left\{(126)/(25),(134)/(25),6,(174)/(25),(206)/(25)\right\}`

So the definite integral is approximated by the sum


`\displaystyle\int_0^2(x^2+5)\,\mathrm dx=\frac25\left((126)/(25)+(134)/(25)+6+(174)/(25)+(206)/(25)\right)=(316)/(25)=12.64`

Compare to the actual value of the integral,


`\displaystyle\int_0^2(x^2+5)\,\mathrm dx=\frac{38}3\approx12.67`