The Meaning of Integration

Integration can be used to find area under a curve. The value of the integral is an area if the integrand is always nonnegative on an interval.

If f(x) ≥ 0 on [ a , b ] then (∫_a^b)f(x) dx ≥ 0.

Find the area of the region bounded by y = (x^3) and the  lines x = 2 and x = 3

Integrate[x^3, {x, 2, 3}]

Integrals can also have a value which is negative or zero.

Two examples of integrals which evaluate to zero. (∫_2^2)(x^3)dx and (∫_0^π)cos xdx.

Integrate[x^3,{x,2,2}]

Two examples of integrals which have negative values. (∫_(-2)^1)(x^3)dx and (∫_2^1)(x^2)dx.

Integrate[x^3,{x,-2,1}]

Integrals can be evaluated by using the geometric interpretation of the integral to simplify the calculations in some cases.

For example, to evaluate (∫_(-2)^(   3)) |x|dx, one would look at the graph of |x|.

Plot[{Abs[x],y=0},{x,-2,3},PlotRange->Automatic, Filling-> Axis]

Now add the areas of the two isosceles right triangles to evaluate the integral.
(∫_(-2)^(   3)) |x|dx= (1/2)(2)(2) + (1/2)(3)(3) =(13/2).

The integral can be used to find the average value of a function over an integral.

Find the average value of f(x) = (e^x) on the interval [ 1, 3 ].

Integrate[E^x,{x,1,3}]/(3 - 1)

which is approximately equal  to

NIntegrate[E^x,{x,1,3},WorkingPrecision->30]/(3 - 1)

Symmetry can be used to simplify integrals.

If  f  is an even function then (∫_(-a)^(  a))f(x) dx = 2(∫_0^a)f(x) dx.
If f  is an odd function then (∫_(-a)^(  a))f(x) dx = 0.
Find (∫_(-1)^(  1))(x^2)+(x^3) dx.  First note that (x^2)is an even function and (x^3) is an odd function, so that
(∫_(-1)^(  1))(x^2)+(x^3) dx= 2(∫_0^(  1))(x^2) dx= 2 (1/3) = 2/3. Check by using Mathematica.

Integrate[x^2+x^3,{x,-1,1}]

Integration can be used to accumulate a rate of change.

If water is leaking from a pool at a rate of r(t) = 100(e^(-2t)) gallons per hour, where t is the time in hours. How many gallons will have leaked out after 24 hours?

Integrate[100E^(-2t), {t,0,24}]

To find a numerical value of an exact numerical answer in Mathematica use N[expression]. So the approximate answer is

N[50-(50/(E^48))]

An integral can be used to find the value of a function at a point.

If the derivative is given as f'(x) and f(a) is known, then the Fundamental Theorem of Calculus says that f(b) - f(a) = (∫_a^b)f'(x)dx, so that f(b) = f(a) +(∫_a^b)f'(x)dx.

Find the approximate value of y at x = 3 , if (dy/dx) = (Sin[x])^3 and y= 2 when x = 1.

2 +NIntegrate[(Sin[x])^3,{x,1,3}, WorkingPrecision->20]

An Integral can be used to define a new function.

One of the most notable examples is the new function, ln(x) for logarithms with base e. 
The definition of ln(x) is  (∫_1^x)(1/t)dt. Then ln(2) = (∫_1^2)(1/t)dt and (d/dx)ln(x) = (d/dx)(∫_1^x)(1/t)dt = (1/x).

Let g(x) =(∫_0^x)cos (t^3)dt. What is the maximum value of g on the interval [ 0, 2 ] ?
Note g(0) = 0 and g'(x) = Cos[x^3]. Since Cos (x) is zero at (π/2), to find the zero of g'(x) take the cube root of  (π/2)

N[(Pi/2)^(1/3)]

or try to find the root of Cos[x^3] = 0. Avoid 0 as a starting point since there is a horizontal tangent line at x = 0.

FindRoot[Cos[x^3]==0,{x,.5,1.5}]

We know that g(0) = 0. Now find the value of g at this point and the value of g at 2.

NIntegrate[Cos[x^3],{x,0,1.1624473515096265}]

NIntegrate[Cos[x^3],{x,0,2}]

Since  0.979022 is the largest of  the three values, it is the maximum on the interval. Plot the graph to verify the answer.

Plot[{Evaluate[NIntegrate[Cos[(x^3)],{x,0,t}]]},{t,0,2}]

The integral is an infinite sum of products. (Riemann Sum)

Find (lim_(n->∞))  ((  n)/( (n^2)))+(n/(((n+1))^2))+ ⋯ + (n/(((2n))^2))

Note that the sum can be rewritten as a product by multiplying by  = (n/n).
This gives the equivalent problem:
 (lim_(n->∞))[ (((((  n)/( n+1))))^2)+ (((((  n)/( n+2))))^2)+ ⋯ +  (((((  n)/(2n))))^2)](1/n)= (lim_(n->∞))(∑_(k=1)^n)((((1/(1+(( k)/n)))))^2)(1/n)= (∫_1^2)(1/(x^2))dx

Integrate[1/x^2,{x,1,2}]

so  we  find   (lim_(n->∞))  ((  n)/( (n^2)))+(n/(((n+1))^2))+ ⋯ + (n/(((2n))^2))=(1/2)

1.  Find the area of the region bounded by x = 1, x = 2, f(x) = (1/(x^3)) and the x-axis.

2.  Evaluate the following: a) (∫_1^(  3))(x-2)dx        b)  (∫_3^0)(x-2)dx        c) (∫_( 0)^(2π))cos xdx     d) (∫_π^0)sin xdx

3.  Use the geometric interpretation of the integral to evaluate (∫_(-1)^(  3))|x-2|dx.

4.  Use the geometric interpretation of the integral to evaluate (∫_0^(  2))(Sqrt[4-(x^2)])dx.

5. What is the average value of  f(x) = x + sin x  on the interval [ 0, π]?

6. Use symmetry to evaluate (∫_(-π)^(  π))(sin x + x +  (x^3) -cos x + 3) dx.

7. If the velocity of an object is v(t) = -16t + 56 and the object is 5 feet off the ground at t = 0, then how high 
      will the object be seven seconds from now?

8. Wine is leaking from a vat at the rate of R(t) = 1000 (e^(-0.1t)) gallons per hour, where t  is measured in hours. 
      How much wine has leaked out of the vat after 12 hours?

9. If  f(0) = 1 and f'(x) = tan((x^3)), then what is f(1)?

10. If  g(x) = (∫_0^x)sin ((t^2))dt on the closed interval [ 0, 3 ], then for what value of x does  g have a local minimum
      and for what value of x does  g have a local maximum? Plot the graph to verify your results.

11. Find (lim_(n->∞ )) [((n^2)/(((n+1))^3))+((n^2)/(((n+2))^3))+⋯+((n^2)/(((2n))^3))]. Hint: Divide by (n^3)  and factor out the (1/n) to form a Riemann sum.