Integral Review
\int4x^2+x+3\ dx
4/3x^3+x^2/2+3x+C
Use the Fundamental Theorem of Calculus to evaluate the following integral:
∫_0^8(3+x)dx
56
∫x^2/(sqrt(x^3+3))\ dx
2/3 sqrt(x^3+3)+C
Use a right endpoint rectangular model to estimate the area under the curves below using 6 rectangles of equal width:
f(x)=x^2, x = 0 and x = 3
11.375
Find the volume: The solid lies between planes perpendicular to the x-axis at x= -4 and x= 4. The cross sections perpendicular to the x-axis are circular disks whose diameters run from the parabola y = x² to the parabola y = 32 - x²
16384/15 * π
\int2x-9sin(x) dx=
x^2+9cos(x)+C
What is x2 + 9cosx+C
Use the Fundamental Theorem of Calculus to evaluate the following integral:
∫_-1^1(4x^3-2x)\ dx
0
∫x(1-3x^2)^4\ dx
-1/30(1-3x^2)^5 +C
Find the total area of the region between:
y= 2x + 7, y=0, x = 1 , x = 5.
52
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
y = \sqrtx, y = 0, x = 0, x = 9
81/2 * π
\intt+e^t \ dt
t^2/2+e^t+C
Use the Fundamental Theorem of Calculus to evaluate the following:
d/dx[∫_0^(x^5)sin(t)\ dt]
5x^4 * sin(x^5)
∫(sinx)^3cosx\ dx
1/4(sinx)^4+C
Find the area of the total region enclosed by the curves:
f(x) = x³+x²-6x,g(x) = 6x
937/12
Find the volume of the solid generated by revolving the region bounded by the curves about the x-axis.
y = -6x + 12, y = 6x, x = 0
72π
\int10/x\ dx
10ln|x| + C
Use the Fundamental Theorem of Calculus to evaluate:
∫_1^6(3/x)\ dx
3ln(6)
∫xe^(-3x^2)\ dx
-1/6e^(-3x^2)+C
Find the area enclosed by the curves:
y = 2x - x², y = 2x - 4
32/3
Find the volume of the solid generated by revolving the region about the y-axis. The region in the first quadrant bounded on the left by y = x³, on the right by the line x = 4, and below by the x-axis.
2048/5 * π
\int6(5^x)\ dx
6/ln(5)5^x + C
Use the Fundamental Theorem of Calculus to evaluate:
d/dx[∫_-2^2t(t^2-6)dt]
0
∫(x^2 + 16)^(1/2]/(x^4)\ dx
−[(x^2+16)^(3/2)]/(48x^3)+C
Find the area enclosed by the curves:
y = x and y = x^2
1/6
The region shown is to be revolved about the x-axis to generate a solid. Which of the methods (disk/washer, shell) could you use to find the volume of the solid? How many integrals would be required in each case?
Washer+Disk (2 integrals), Shell (1 integral)