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definite integral:
(x+2)dx
2
1
find antiderivative
plug in 2 minus plug in 1
area between x =1, x=2, f(x) = x+2 and the x axis
for a function y = f(x0, nonnegative and continuous on [a,b], the area under the curve is given by
A = integral(lowa, highb) f(x) f(x) dx
fundamental theory of calculus is how to find this
generally its plugin b - plugin a
integral from 1 to 1 is 0 a to a is 0 2 to 2 is 0 etc
no area between 1 and 1.
evaluate
integral(2, 1) 2xdx
antiderivative (x^2) 2^2 - 1^2 4 - 1 3
3x^2 - 4x + 1
x^3 - 2x^2 + x
now we find at 4 and 1
4^3 - 2(4)^2 + 4 64 - 2(16) + 4 64 - 32 + 4 64 - 28 36
1^3 - 2(1)^2 + 1 1 - 2 + 1 0
36 - 0 36
(1,0) e^x dx
e
e^1 = e e^0 = 1
e - 1
for a function y = f(x) continuous on the interval a,b the average value is 1/(b-a) integral(b,a) f(x)dx
find average value of f(x) = x^2+2 on the interval 0,3
1/(3-0) * integral(3,0) (x^2+2)dx 1/(3-0) * integral(3,0) ((x^3)/3+2x) 1/3 * ((3^3)/3 + 2(3) - (0^3/3 + 2(0)) 1/3 * (9/3 + 2(3) - 0) 1/3 (15) 5
-2 / x^3 -2x^-3 antiderivative -2x^-2 / -2 x
now we solve f(4) - f(3) (4)^-2 - (3)^-2 1/4^2 - 1/3^2 1/16 - 1/9 9/144 - 16/144 -7/144
-7/x
-7x^-2 -7x^-1 / -1 7/x
f(6) - f(3)
7/6 - 7/3
7/6 - 14/6 -7/6
-7x+2 -(7/2)x^2 + 2x
f(-4) - f(-5) -(7/2)(-4)^2 + 2(-4) -7/2 * 16 - 8 -56 - 8 -64
-7/2 * -5^2 + 2(-5) -7/2 * 25 - 10 -87.5 - 10 -97.5
-64 + 97.5 33.5
5+5e^x 5x + 5e
f(5) - f(4) 5(5) + 5e^(5) 25 + 5e
5(4) + 5e^4 20 + 5e
25 + 5e^5 - 20 - 5e^4 5 + 5e^5 - 5e^4 474.075045347 474.08
3/x
3x^-1 3 * 1/x 3ln|x|
f(19) - f(16) 0.51555077078 .52
3(3x-8)^2 u = 3x-8 du = 3
u^3/3
(3x-8)^3 / 3
f(4) - f(-1) (3(4)-8)^3 / 3 (12-8)^3 / 3 4^3 / 3 64/3
(3(-1)-8)^3 / 3 (-3-8)^3 / 3 (-11)^3/3 -1331/3
64/3 - -1331/3 64/3 + 1331/3 1395/3 465
-24x(3x^2-5)^3 u = 3x^2-5 du = 6x -4 integral (3x^2-5)^3 6xdx
-4 u^3 -4 u^4/4
-4 (3x^2-5)^4 / 4 -(3x^2-5)
now solve nation
f(3) - f(1)
-(3(3)^2-5)^4 -(3(9)-5)^4 -(27-5)^4 -(22)^4 -234256
-(3(1)^2-5)^4 -(3-5)^4 -(-2)^4 -16
-234256 + 16 -234240
5x^2 + 6 5x^3/3 + 6x
f(3)-f(1) 5(3)^3/3 + 6(3) 5(3)^2 + 6(3) 5(9) + 18 45 + 18 63
5(1)^3/3 + 6(1) 5/3 + 6 23/3
63 - 23/3 189/3 - 23/3 166/3
166/3 * (1/b-a) 166/3 * (1/3-1) 166/3 * 1/2 83/3
180 - 180e^(-.2x)
180x + integrate -180e^(-.2x)
-180e^(-.2x) exponent u = -.2x du = -.2
900 e^(-.2x) * -.2dx
900 e^(-.2x)
180x + 900e^(-.2x)
f(2) - f(0)
180(2) + 900e^(-.2(2)) 360 + 900e^(-.4)
180(0) + 900e^(-.2(0)) 900e^0 900
360 - 900e^(-.4) - 900 -540 + 900e^(-.4)
(1/2) ( -540-900e^(-.4) -270 + 450e^(-.4)
1/(7x+6)
u sub, demoninator
u = 7x+6 du = 7
1/7x+6
1/7 * integrate: 7/7x+6
1/7 ln|u| 1/7ln|7x+6| 1/7ln|7(5)+6| 0.530510295243 1/7ln|7(1)+6| 0.36642133678
f(5) - f(1) = 0.164088958463
.16
4 + 2e^(-1.4x) 4x + integral 2e^(-1.4x)
2e^(-1.4x)
u = -1.4x du = -1.4
-2/1.4 e^(-1.4x) -1.4dx
4x + -2/1.4 * e^(-1.4x)
f(6)-f(3) 12.0211010136 12.02
8xe^(-x^2+7)
inside exponent denominator exponent u = -x^2 + 7 du = -2x
e^(-x^2+7) * 8xdx
need to get 8x to -2x
-4 e^(-x^-2+7) * -2xdx
-4 e^(-x^2+7)
now f(3) - f(-1)
1613.17383284 1613.17
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