Evaluate the following integral by interpreting it in terms of areas:
Value of integral

Ask by Park Stephens. in the United States

Mar 27,2025

Question

Evaluate the following integral by interpreting it in terms of areas: Value of integral $$ =\square \int_{0}^{3}\left(\frac{1}{2} x-1\right) d x $$

UpStudy AI · Accepted Answer

Calculate the integral $$ \int_{0}^{3}(\frac{1}{2}x-1)dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int_{0}^{3} \left(\frac{1}{2}x-1\right) dx$$
- step1: Evaluate the integral:
  $$\int \left(\frac{1}{2}x-1\right) dx$$
- step2: Use properties of integrals:
  $$\int \frac{1}{2}x dx+\int -1 dx$$
- step3: Evaluate the integral:
  $$\frac{x^{2}}{4}+\int -1 dx$$
- step4: Evaluate the integral:
  $$\frac{x^{2}}{4}-x$$
- step5: Return the limits:
  $$\left(\frac{x^{2}}{4}-x\right)\bigg |_{0}^{3}$$
- step6: Calculate the value:
  $$-\frac{3}{4}$$
The value of the integral $$ \int_{0}^{3}(\frac{1}{2}x-1)dx $$ is $$-\frac{3}{4}$$ or $$-0.75$$.
The value of the integral is $$-0.75$$.

Evaluate the following integral by interpreting it in terms of areas:
Value of integral

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