Determine the area of the shaded region in the following figures.

검증된 단계별 안내

Step 1: Identify the two curves that bound the shaded region. The upper curve is y =

\frac{6}{x^{2} + 1}

, and the lower curve is y =

3 x^{2}

Step 2: Determine the points of intersection between the two curves by setting

\frac{6}{x^{2} + 1}

equal to

3 x^{2}

. Solve for x to find the limits of integration.

Step 3: Set up the integral to calculate the area of the shaded region. The area is given by the integral of the difference between the upper curve and the lower curve:

\int_{a}^{b} \frac{6}{x^{2} + 1} - 3 x^{2} dx

, where a and b are the x-values of the intersection points.

Step 4: Simplify the integrand. Combine the terms inside the integral:

\frac{6}{x^{2} + 1} - 3 x^{2}

. This will give you the function to integrate.

Step 5: Evaluate the integral over the interval [a, b] to find the area of the shaded region. Use the fundamental theorem of calculus to compute the definite integral.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definite Integral

The definite integral is a fundamental concept in calculus that represents the signed area under a curve between two points on the x-axis. It is calculated using the integral symbol and provides a way to quantify the area of regions bounded by curves. In this context, the area of the shaded region between the two curves can be found by integrating the upper curve minus the lower curve over the interval where they intersect.

Finding Points of Intersection

To determine the area between two curves, it is essential to find their points of intersection. These points are where the two curves meet, and they define the limits of integration for calculating the area. By setting the equations of the curves equal to each other, we can solve for the x-values at which they intersect, which are crucial for setting up the definite integral.

Area Between Curves

The area between two curves is calculated by integrating the difference of the functions that define the curves over the interval defined by their points of intersection. This area can be expressed mathematically as ∫[a, b] (f(x) - g(x)) dx, where f(x) is the upper curve and g(x) is the lower curve. Understanding this concept is vital for solving problems that involve shaded regions between curves.