Math Problem Statement
Solution
To determine the area of the figure bounded by the lines given:
we need to set up the definite integral of from to .
The integral to calculate the area is:
To solve this, let's use the substitution method:
Let , then or .
Thus, the integral becomes:
This can be integrated using the power rule:
[ \frac{1}{2} \int_{0}^{1} u^{-1/6} , du = \frac{1}{2} \left[ \frac{u^{5/6}}{5/6} \right]{0}^{1} = \frac{1}{2} \left( \frac{6}{5} \left[ u^{5/6} \right]{0}^{1} \right) = \frac{1}{2} \cdot \frac{6}{5} \left( 1^{5/6} - 0^{5/6} \right) = \frac{1}{2} \cdot \frac{6}{5} = \frac{3}{5} = 0.6 ]
Therefore, the area of the figure is 0.6.
Would you like any further details or explanations?
Here are some questions you might want to ask:
- How do you find the area under a curve?
- What is the power rule for integration?
- How does substitution work in integration?
- What are definite integrals?
- Can you show another example of finding an area using integrals?
Tip: When integrating, always double-check your substitutions and limits to ensure accuracy.
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Math Problem Analysis
Mathematical Concepts
Calculus
Integral calculus
Definite integrals
Formulas
Integral of u^(1/3)
Theorems
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Suitable Grade Level
Advanced High School