Use a geometry formula to find the exact value of the definite integral.
4x dx on the interval [0,b]

Question

Use a geometry formula to find the exact value of the definite integral.  4x dx on the interval [0,b]

Theirl1972 · Accepted Answer

Step 1
The given function is 4x with the interval [0,b].
Step 2
Find the exact value of the given integral as follows.

\int_{0}^{b} 4 x d x = \int_{0}^{b} 4 x d x

= 4 \int_{0}^{b} x d x

= 4 {[\frac{x^{1 + 1}}{1 + 1}]}_{0}^{b}

= 4 {[\frac{x^{2}}{2}]}_{0}^{b}

= 4 [\frac{{(b)}^{2}}{2} - \frac{{(0)}^{2}}{2}]

Step 3
On further simplification,

\int_{0}^{b} 4 x d x = 4 [\frac{b^{2}}{2} - 0]

= 4 (\frac{b^{2}}{2})

= 2 b^{2}

Step 4
Answer:
The exact value of the given definite integral is

b^{2}

.

Charles Clute · Accepted Answer

Step 1: If f(x) is a continuous function from a to b, and if F(x) is its integral, then:

\int_{a}^{b} f (x) d x = F (x) ∣_{a}^{b} = F (b) - F (a)

Step 2: In this case, f(x)=4x. Find its integral.

2 x^{2} ∣_{0}^{b}

Step 3: Since

F (x) ∣_{a}^{b} = F (b) - F (a)

, expand the above into F(b)−F(0):

2 b^{2} - 2 \times 0^{2}

Step 4: Simplify.

2 b^{2}

Use a geometry formula to find the exact value of