Calculus 1: Antiderivatives, Riemann Sums, and Integrals
88 videos • 2,802 views • by Math Problems Solved Craig Faulhaber
1
Find the family of antiderivatives of the function f(x) = 5/x^2.5 + x^10
Math Problems Solved Craig Faulhaber
Download
2
Find the family of antiderivatives of the function f(x) = 4x^3 + x^2 - 2
Math Problems Solved Craig Faulhaber
Download
3
Find the family of antiderivatives of the function f(x) = -4.5 / sqrt(x^7)
Math Problems Solved Craig Faulhaber
Download
4
Find the family of antiderivatives of the function f(x) = x^(5/2) + 7x^(1/2)
Math Problems Solved Craig Faulhaber
Download
5
Find the family of antiderivatives of the function f(x) = sinx + e^(2x)
Math Problems Solved Craig Faulhaber
Download
6
Find the family of antiderivatives of the function f(x) = cos(2x) + 3e^(-4x)
Math Problems Solved Craig Faulhaber
Download
7
Find the family of antiderivatives of the function f(x) = -3sec(x) ( sec(x) - tan(x) ). F(x)=___
Math Problems Solved Craig Faulhaber
Download
8
Evaluate the integral int ( (.5x^7+4.5)/x^7 ) dx
Math Problems Solved Craig Faulhaber
Download
9
Find the family of antiderivatives of f with the given condition f(x) = e^7x - 1.5x^2 and F(0)=8
Math Problems Solved Craig Faulhaber
Download
10
Find a function satisfying the conditions f'''(x)=cosx, f(0)=8, f'(0)=0, f''(0)=-1
Math Problems Solved Craig Faulhaber
Download
11
Find a function satisfying the conditions f'''(x)=27cos(3x), f(0)=-8, f'(0)=1, f''(0)=3
Math Problems Solved Craig Faulhaber
Download
12
Use summation properties and formulas to evaluate the sum, Sum_(j=1)^26 ( j^2 + 7j )
Math Problems Solved Craig Faulhaber
Download
13
Use summation properties and formulas to evaluate the sum, Sum_(k=1)^23 ( (5k)^2 + 8k )
Math Problems Solved Craig Faulhaber
Download
14
Evaluate the sum: Sum_(k=100)^200 k^3
Math Problems Solved Craig Faulhaber
Download
15
Given the function f(x)=1/(x^2+4), compute the left-endpoint Riemann sum using n=2 on the interval
Math Problems Solved Craig Faulhaber
Download
16
Estimate the area under the curve by computing the left Riemann sum L_6
Math Problems Solved Craig Faulhaber
Download
17
Estimate the area under the curve by computing the left Riemann sum L_8
Math Problems Solved Craig Faulhaber
Download
18
Express the limit of S_n as an integral over [-2,2]. S_n = Sum_(i=1)^n (6(x_i)^4-4(x_i)^5) delta x
Math Problems Solved Craig Faulhaber
Download
19
Express the limit of R_n as a definite integral, R_n = 1/n Sum_(i=1)^n ( 5 + i (1/n) )
Math Problems Solved Craig Faulhaber
Download
20
Express the limit of L_n as a definite integral, L_n = 5/n Sum_(i=1)^n ( 9 + (i - 1) (5/n) )
Math Problems Solved Craig Faulhaber
Download
21
Evaluate the integral of the function graphed using the formula for areas of triangles
Math Problems Solved Craig Faulhaber
Download
22
Evaluate the integral of the function graphed using the formula for areas of triangles and circles
Math Problems Solved Craig Faulhaber
Download
23
Evaluate the integral below by interpreting it in terms of areas int_-4^4 ( sqrt(16-x^2) ) dx
Math Problems Solved Craig Faulhaber
Download
24
Evaluate the integral int_0^10 sqrt( 25 - (x-5)^2 ) using area formulas
Math Problems Solved Craig Faulhaber
Download
25
int_4^17 f(x) - int_4^9 f(x) = int_a^b f(x) where a=___ and b=___
Math Problems Solved Craig Faulhaber
Download
26
Evaluate int_3^3 (x^3 cos(6x)e^(6x) + 50 ) dx
Math Problems Solved Craig Faulhaber
Download
27
Let f(x) = integral from -1 to infty of t^7 dt. Find f'(x) and f'(3)
Math Problems Solved Craig Faulhaber
Download
28
Use the Fundamental Theorem of Calculus, to find the following: d/dx ( int_5^(3x) e^cost dt )
Math Problems Solved Craig Faulhaber
Download
29
Use the Fundamental Theorem of Calculus to find d/dx (int_4^(4x^3) t dt )
Math Problems Solved Craig Faulhaber
Download
30
Use the Fundamental Theorem of Calculus to find d/dx (int_cos(4x)^1 sqrt(1 - t^2) dt )
Math Problems Solved Craig Faulhaber
Download
31
If F(x) = (int_(x^3)^(x^5) (2t - 1)^3 dt ), then find F'(x) using the Fundamental Theorem Part 1
Math Problems Solved Craig Faulhaber
Download
32
The graph of y = int_0^x f(t) dt is shown here. Over which intervals is f positive, negative, zero?
Math Problems Solved Craig Faulhaber
Download
33
Use part 2 of the Fundamental Theorem of Calculus to integrate int_4^12 ( 1/x^2 ) dx
Math Problems Solved Craig Faulhaber
Download
34
Integrate 3sqrt(x) from x=4 to x=18
Math Problems Solved Craig Faulhaber
Download
35
Integrate int( 18x^2 - 8x + 7 ) from x = 4 to 7
Math Problems Solved Craig Faulhaber
Download
36
Integrate int_4^7 ( 5/sqrt(x) ) dx
Math Problems Solved Craig Faulhaber
Download
37
Evaluate int_4^12 ( 8t^(9/2) - 4 t^(7/2) ) dt using the Fundamental Theorem of Calculus Part 2
Math Problems Solved Craig Faulhaber
Download
38
Evaluate int_4^12 ( 8t^(9/2) - 4 t^(7/2) ) dt using the Fundamental Theorem of Calculus Part 2
Math Problems Solved Craig Faulhaber
Download
39
Evaluate int_10^12 ( ( 10-sqrt(t) )/t^2 ) dt using the Fundamental Theorem of Calculus Part 2
Math Problems Solved Craig Faulhaber
Download
40
Evaluate int_0^pi/3 ( -8sec(theta)tan(theta) ) d(theta) using the Fundamental Theorem of Calculus
Math Problems Solved Craig Faulhaber
Download
41
Integral from 0 to 1 of ( 4e^x - 4sinx ) dx
Math Problems Solved Craig Faulhaber
Download
42
Integral from 1 to sqrt(7) of 6/(1+x^2) dx
Math Problems Solved Craig Faulhaber
Download
43
Evaluate int_0^2 f(x) dx where f(x) is the piecewise function 5x^7 for x from 0 to 1, 4x^3 for x fro
Math Problems Solved Craig Faulhaber
Download
44
Use the evaluation theorem to express int_3^x (-3e^(-2t) dt ) as a function F(x)
Math Problems Solved Craig Faulhaber
Download
45
Identify the roots of the integrand in int_-19^5 |t^2+14t+45| dt to remove the absolute value, then
Math Problems Solved Craig Faulhaber
Download
46
Evaluate the following definite integral int_0^pi/2 ( 13x - 4sinx ) dx
Math Problems Solved Craig Faulhaber
Download
47
Evaluate the following indefinite integral int (9/sqrt[3](x) - 7 sqrt[3](x) ) dx
Math Problems Solved Craig Faulhaber
Download
48
Evaluate the following indefinite integral int ( 7x-2 )/x^2 dx
Math Problems Solved Craig Faulhaber
Download
49
Evaluate the following indefinite integral int ( e^(x/2)/10 - e^(5x) ) dx
Math Problems Solved Craig Faulhaber
Download
50
A car drives in such a way that its velocity is v(t) = t^(1/2)+1. Find the average velocity between
Math Problems Solved Craig Faulhaber
Download
51
Suppose that a particle moves along a straight line with velocity v(t)=4-2t. Find the displacement a
Math Problems Solved Craig Faulhaber
Download
52
A car moves with acceleration a(t)=t-5/2. Find the velocity, displacement and distance traveled give
Math Problems Solved Craig Faulhaber
Download
53
A ball is thrown upward from a height 3.5 m at an initial speed of 50 m/s. Solve for the velocity an
Math Problems Solved Craig Faulhaber
Download
54
Use the substitution u=3x-2 to evaluate int(3x-2)^-9 dx.
Math Problems Solved Craig Faulhaber
Download
55
Use the substitution u=4x^2+3 to find the integral int ( 2x / sqrt(4x^2 + 3) ) dx
Math Problems Solved Craig Faulhaber
Download
56
Use a suitable change of variables to evaluate the integral int( 8x(3 - 3x)^5 dx )
Math Problems Solved Craig Faulhaber
Download
57
Use a suitable change of variables (u-substitution) to evaluate the integral int 7sin^2(x)cos(x) dx
Math Problems Solved Craig Faulhaber
Download
58
Integrate 5tsin(36t^2)cos(36t^2) using an appropriate substitution
Math Problems Solved Craig Faulhaber
Download
59
Integrate cos(pi x) from x = 0 to 7 using an appropriate substitution
Math Problems Solved Craig Faulhaber
Download
60
Integrate cosx (1 - cosx)^8 sinx dx using a u-substitution
Math Problems Solved Craig Faulhaber
Download
61
Integrate 7/2 sec^2(x) tan^10(x) dx
Math Problems Solved Craig Faulhaber
Download
62
Integrate 7cos^2(4x)sin(4x)dx from x=0 to x=pi
Math Problems Solved Craig Faulhaber
Download
63
Integrate cos(z)/sin^4(z) dz from z=pi/6 to z=pi/2 using an appropriate substitution
Math Problems Solved Craig Faulhaber
Download
64
Find the integral of 9 sin^(x) cos^3(x) dx
Math Problems Solved Craig Faulhaber
Download
65
Calculate the integral of 1/x^2 sqrt(1+1/x) dx from x=1 to x=4
Math Problems Solved Craig Faulhaber
Download
66
If int_0^4 f(x) dx = -10 then find int_0^2 f(2x) dx
Math Problems Solved Craig Faulhaber
Download
67
Given that int f(x)dx from x=0 to 9 is 4, find the value of int x f(x^2)dx from x=0 to 3
Math Problems Solved Craig Faulhaber
Download
68
Evaluate the integral of x^2 sqrt(4x+5) dx from 0 to 1
Math Problems Solved Craig Faulhaber
Download
69
Evaluate the integral of ( 22 e^(2x) - 8 )dx Bonus: int e^(ax)dx
Math Problems Solved Craig Faulhaber
Download
70
Integrate ( 23^(13x) + 8 ) dx
Math Problems Solved Craig Faulhaber
Download
71
Integrate 10/7 ( 1/x(lnx)^8 ) dx using a rad substitution :)
Math Problems Solved Craig Faulhaber
Download
72
Integral e^( ln(x) )/x with the bonus problem: Integral ln(2x) / (3x) dx
Math Problems Solved Craig Faulhaber
Download
73
Write an integral to represent the area between the curves y=x^2-1 and y=6
Math Problems Solved Craig Faulhaber
Download
74
Calculus POD 52. Find all of the antiderivatives for the following function f(y)=−26/y^27
Math Problems Solved Craig Faulhaber
Download
75
Calculus 1 POD #53 Initial Value Problem
Math Problems Solved Craig Faulhaber
Download
76
Calculus POD 54 Riemann Sums for f(x)=cos(x-pi/4)
Math Problems Solved Craig Faulhaber
Download
77
Calc 1 POD 55 Riemann Sums (Left, Right, Midpoint)
Math Problems Solved Craig Faulhaber
Download
78
Calculus POD 56 Areas and Integral Rules.
Math Problems Solved Craig Faulhaber
Download
79
Calculus 1 POD 57 Evaluate the definite Integral ∫_(1/2)^1 (x^(−3)−8) dx
Math Problems Solved Craig Faulhaber
Download
80
Calculus POD 58 Find the Area Using Integrals
Math Problems Solved Craig Faulhaber
Download
81
Calc 1 POD 59 Average value of a function f(x)=x^3−6x^2+2 on [0,4]
Math Problems Solved Craig Faulhaber
Download
82
Calc POD 60 Average Value! Find the points at which the function f(x)=7−6x equals its average val...
Math Problems Solved Craig Faulhaber
Download
83
Calculus POD 61 Integrate using a u substitution ∫2x(x^2+13)^8 dx
Math Problems Solved Craig Faulhaber
Download
84
Calculus POD 62 Use a change of variables to evaluate the integral ∫(x^6−2x^3 )^4 (x^5−x^2 )dx
Math Problems Solved Craig Faulhaber
Download
85
Calculus POD #63 displacement vs distance
Math Problems Solved Craig Faulhaber
Download
86
Calculus POD 64 Find the position and velocity of an object with the given acceleration, initial...
Math Problems Solved Craig Faulhaber
Download
87
Calculus POD 65 Find the area of the shaded region shown in the graph
Math Problems Solved Craig Faulhaber
Download
88
Calculus POD 66 Find the area of the shaded region shown in the figure between y=7-x and x=(y-4)^2/4
Math Problems Solved Craig Faulhaber
Download