4.5. Evaluating Definite Integrals http://www.ck12.org
Note thatF(a) =∫aaf(x)dx= 0 .Hence we have
f(c) =F(bb−)−a^0 =Fb−(ba),and by our definition ofF(x)we have
f(c) =b−^1 aF(b) =b−^1 a∫b
a f(x)dx.This theorem allows us to find for positive functions a rectangle that has base[a,b]and heightf(c)such that the area
of the rectangle is the same as the area given by∫abf(x)dx.In other words,f(c)is the average function value over
[a,b].
Review Questions
In problems #1–8, use antiderivatives to compute the definite integral.
1.∫ 49 (√^3 x)dx
2.∫ 01 (t−t^2 )dt
3.∫ 25 (√^1 x+√^12 )dx
4.∫ 014 (x^2 − 1 )(x^2 + 1 )dx
5.∫ 28 (^4 x+x^2 +x)dx
6.∫ 24 (e^3 x)dx7.∫ (^14) x+^23 dx
- Find the average value off(x) =√xover[ 1 , 9 ].
- Iffis continuous and∫ 14 f(x)dx= 9 ,show thatftakes on the value 3 at least once on the interval[ 1 , 4 ].
- Your friend states that there is no area under the curve off(x) =sinxon[ 0 , 2 π]since he computed∫ 02 πsinxdx=
0 .Is he correct? Explain your answer.