Solved: "Use the form of the definition of the..."

High School
Calculus and Analysis
Calculus 2
Applications of integrals

David Troyer

Answered question

2021-12-16

Use the form of the definition of the integral given in
Theorem 4 to evaluate the integral.

\int_{0}^{2} (2 x - x^{3}) d x

Answer & Explanation

temzej9

Beginner2021-12-17Added 30 answers

Step 1
To evaluate the integral:

\int_{0}^{2} (2 x - x^{3}) d x

Evaluating the integral.

\int_{0}^{2} (2 x - x^{3}) d x = \int_{0}^{2} 2 x d x - \int_{0}^{2} x^{3} d x

= {(2 \cdot \frac{x^{2}}{2})}_{0}^{2} - {(\frac{x^{4}}{4})}_{0}^{2}

= {(x^{2})}_{0}^{2} - (\frac{2^{2}}{4} - 0)

=(4-0)-(1-0)
=4-1
=3
Step 2
Hence, required answer is 3.

levurdondishav4

Beginner2021-12-18Added 38 answers

\int (2 x - x^{3}) d x

Lets

nick1337

Expert2021-12-28Added 777 answers

We represent the original integral as a sum of tabular integrals:
$\int (2 * x - x^{3}) * d x = \int - x^{3} * d x + \int 2 * x * d x$
$\int (- x^{3}) * d x$
This is a tabular integral:
$\int - x^{3} * d x = - \frac{x^{4}}{4} + C$
$\int 2 * x * d x$
This is a tabular integral:
$\int 2 * x * d x = x^{2} + C$
$\int (2 * x - x^{3}) * d x = - \frac{x^{4}}{4} + x^{2} + C$
Let's calculate the definite integral:
$\int_{0}^{2} (2 * x - x^{3}) * d x = (- \frac{x^{4}}{4} + x^{2})_{0}^{2}$
F(2)=0
F(0)=0
I=0-(0)=0

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