Question

In Exercises 11–14, find the total area of the region between the graph of f and the x-axis.ƒ(x) = 5 - 5x²/³, -1 ≤ x ≤ 8

Accepted Answer

First, understand that the total area between the graph of the function and the x-axis means we need to find the integral of the absolute value of the function over the interval $$-1 \leq x \leq 8. $$This is because areas below the x-axis contribute positively to the total area when taken as absolute values. Next, identify where the function $$f(x) = 5 - 5x^{2/3}$$ intersects the x-axis by solving $$f(x) = 0. $$Set $$5 - 5x^{2/3} = 0$$ and solve for $$x to $$find the points where the graph crosses the x-axis. Once the zeros are found, split the integral into subintervals where the function is either positive or negative. For each subinterval, set up the definite integral of $$|f(x)|$$ which will be either $$f(x) or$$ $$-f(x)$$ depending on the sign of $$f(x) in $$that interval. Write the total area as the sum of the integrals over these subintervals: $$	ext{Area} = \int_{a}^{b} |f(x)| \, dx = \sum \int_{x_i}^{x_{i+1}} |f(x)| \, dx$$, where $$x_i$$ are the zeros and the endpoints $$-1$$ and $$8. $$Finally, compute each integral separately by integrating $$f(x) or$$ $$-f(x) as $$appropriate, using the power rule for integration on $$x^{2/3}$$, and then sum the absolute values of these integrals to get the total area.

In Exercises 11–14, find the total area of the region between the graph of f and the x-axis.
ƒ(x) = 5 - 5x²/³, -1 ≤ x ≤ 8

Verified video answer for a similar problem:

Key Concepts

Definite Integral and Area Under a Curve

Identifying Points Where the Function Crosses the x-axis

Handling Functions with Fractional Exponents