Definite Integral

In other words:

• The definite integral of \(f(x)\) from \(x = a\) to \(x = b\) is the area under the curve \(y = f(x)\) from \(x = a\) to \(x = b\).
• This is computed using limits of Riemann sums.
• Today: find a better method for computing these areas.

If \(f(x)\) is integrable on \([a, b]\), we can define:

\[ F(x) = \int_a^x f(t) dt \]

Idea

• Pick \(x\) in \([a, b\)].
• Find area from \(a\) to \(x\) under the curve.
• That’s the value of \(F(x)\).
• Q: What is \(F(a)\)? Area of a straight, vertical line?

Example

Using geometry: let \(F(x) = \int\limits_0^x 2t dt\). Find an expression for \(F(x)\).

Solution

• Triangle: \(A = \frac{1}{2} bh\)
• Base = \(x\)
• Height = \(2x\)
• \(A = \frac{1}{2} x \cdot (2x) = x^2\)

Exercise

• \(\int\limits_2^x 2t dt\)
• \(\int\limits_3^x 2t dt\)

Hint

Area of a trapezoid: \(\frac{1}{2} a(b_1 + b_2)\)

Solution

\[ \begin{align} \int_2^x 2t dt &= \frac{1}{2}(x-2)(4+2x) \\ &= (x-2)(2+x) \\ &= x^2 - 4 \end{align} \]

Solution

\[ \begin{align} \int\limits_3^x 2t dt &= \frac{1}{2}(x-3)(6+2x) \\ &= (x-2)(3+x) \\ &= x^2 - 9 \end{align} \]

General pattern? \(\int\limits_a^x 2t dt = x^2 - a^2\)

Notice: \((x^2 - a^2)^\prime = 2x\)!

Theorem: If \(f(x)\) is continuous on \([a, b]\) and

\[ F(x) = \int_a^x f(t) dt\]

then \(F^\prime(x) = f(x)\)

“Derivative of an integral cancels out.”

Idea

\(F(x+h) - F(x)\)?

Idea

\[F(x+h) - F(x) = \int_x^{x+h} f(t) dt\]

• As \(h \rightarrow 0\)?
• \(F(x+h) - F(x) \approx f(x)h\) (Riemann)
• So \(\dfrac{F(x+h)-F(x)}{h}\)?

Upshot?

• Every continuous function on \([a, b]\) has an antiderivative!
• Ex: \(\frac{d}{dx} \int\limits_0^x \frac{1}{t+1} dt = \frac{1}{x+1}\)

Example:

\(\frac{d}{dx} \int\limits_0^{x^2} \sin(t) dt\)? Chain rule:

• \(F(x) = \int\limits_0^x \sin(t) dt\)
• \(g(x) = x^2\)
• \((F \circ g)^\prime(x)\)?

\(\frac{d}{dx} \int\limits_0^{x^2} \sin(t) dt\):

• \(F^\prime(g(x)) g^\prime(x)\)
• \(F^\prime(x) = \sin(x)\)
• \(g^\prime(x) = 2x\)
• So: \(2x \sin(x^2)\)

Exercises

Find the derivatives of the following functions:

1. \(F(x) = \int\limits_{-2}^{2x+1} t^2 dt\)
2. \(G(x) = \int\limits_0^{x^3} t dt\)
3. \(H(x) = \int\limits_{x^2}^{1} t dt\)

Solution 1

\[ \begin{align} F_1(x) &= \int_{-2}^{x} t^2 dt \\ g(x) &= 2x + 1 \\ F(x) &= (F_1 \circ g)(x) \\ F^\prime(x) &= F_1^\prime(g(x)) g^\prime(x) \\ &= (2x+1)^2 \cdot 2 \end{align} \]

Solution 2

\[ \begin{align} G_1(x) &= \int_{0}^{x} t dt \\ g(x) &= x^3 \\ G(x) &= (G_1 \circ g)(x) \\ G^\prime(x) &= G_1^\prime(g(x)) g^\prime(x) \\ &= x^3 \cdot (3x^2) \end{align} \]

Solution 3

• Trickier.
• \(F(x) = \int\limits_1^x t dt\)
• \(-F(x) = \int\limits_x^1 t dt\)
• \(g(x) = x^2\)
• \((-F \circ g)^\prime(x)\)?
• \(= (-x^2)(2x) = -2x^3\)

FTC Part 2

Theorem: If \(F(x)\) is an antiderivative of \(f(x)\), defined on \([a, b]\), then

\[\int_a^b f(x) dx = F(b) - F(a)\]

Example

\[\int_0^1 x^2 dx\]

• Antiderivative?
• \(F(x) = \frac{x^3}{3}\)
• \(F(1) - F(0) = \frac{1}{3} - 0 = \frac{1}{3}\)

Example

\[\int_{-\pi}^{\pi} \sin(x) dx\]

• Antiderivative?
• \(F(x) = -\cos(x)\)
• \(F(\pi) = -\cos(\pi) = 1\)
• \(F(-\pi) = -\cos(-\pi) = 1\)
• \(F(1) - F(0) = 0\)

Exercises

Evaluate the following definite integrals (using the fundamental theorem of calculus):

1. \(\int\limits_0^{\pi} \cos(x) dx\)
2. \(\int\limits_1^{e} \frac{1}{x} dx\)
3. \(\int\limits_0^{2\pi} \sin(x) dx\)