5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 15:57

252 STEP 4. Review the Knowledge You Need to Score High

Iff(x),g(x), andh(x) are integrable on [a,b], then

3.

∫a

a

f(x)dx= 0

4.

∫b

a

f(x)dx=−

∫a

b

f(x)

5.

∫b

a

Cf(x)dx=C

∫b

a

f(x)dxwhenCis a constant.

6.

∫b

a

[f(x)±g(x)]dx=

∫b

a

f(x)dx±

∫b

a

g(x)dx

7.

∫b

a

f(x)dx≥0 provided f(x)≥0on[a,b].

8.

∫b

a

f(x)dx≥

∫b

a

g(x)dxprovided f(x)≥g(x)on[a,b].

9.

∣∣

∣∣

∫b

a

f(x)dx

∣∣

∣∣≤

∫b

a

∣

∣f(x)

∣

∣dx

10.

∫b

a

g(x)dx≤

∫b

a

f(x)dx≤

∫b

a

h(x)dx; providedg(x)≤f(x)≤h(x)on[a,b].

11. m(b−a)≤

∫b

a

f(x)dx≤M(b−a); providedm≤ f(x)≤Mon [a,b].

12.

∫c

a

f(x)dx=

∫b

a

f(x)dx+

∫c

b

f(x)dx; providedf(x) is integrable on an interval

containinga,b,c.

Examples

1.

∫π

π

cosxdx= 0

2.

∫ 5

1

x^4 dx=−

∫ 1

5

x^4 dx

3.

∫ 7

− 2

5 x^2 dx= 5

∫ 7

− 2

x^2 dx

4.

∫ 4

0

(

x^3 − 2 x+ 1

)

dx=

∫ 4

0

x^3 dx− 2

∫ 4

0

xdx+

∫ 4

0

1 dx

5.

∫ 5

1

√

xdx=

∫ 3

1

√

xdx+

∫ 5

3

√

xdx

Or

∫ 3

1

√

xdx=

∫ 5

1

√

xdx+

∫ 3

5

√

xdx

∫c

a

=

∫b

a

+

∫c

b

a, b, cdo not have to be arranged from smallest to largest.