Find the area of the region described in the following exercises.

The region bounded by x=y(y−1) and y=x/3

검증된 단계별 안내

Step 1: Identify the curves that bound the region. The first curve is given as \( x = y(y - 1) \), which can be rewritten as \( x = y^2 - y \). The second curve is \( y = \frac{x}{3} \).

Step 2: Find the points of intersection between the two curves. To do this, substitute \( y = \frac{x}{3} \) into \( x = y^2 - y \). Replace \( y \) with \( \frac{x}{3} \) to get \( x = \left(\frac{x}{3}\right)^2 - \frac{x}{3} \). Solve this equation for \( x \) to determine the limits of integration.

Step 3: Set up the integral to calculate the area. The region is bounded vertically, so express the area as \( \int_{x_1}^{x_2} \left( \text{top curve} - \text{bottom curve} \right) dx \). Here, the top curve is \( y = \frac{x}{3} \) and the bottom curve is \( y = \sqrt{x + x^2} \) (after solving for \( y \) in terms of \( x \)).

Step 4: Determine the limits of integration \( x_1 \) and \( x_2 \) from the points of intersection found in Step 2. These values will serve as the bounds for the integral.

Step 5: Evaluate the integral \( \int_{x_1}^{x_2} \left( \frac{x}{3} - \sqrt{x + x^2} \right) dx \) to find the area of the region. Break the integral into manageable parts if necessary, and simplify before solving.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Defining the Region

To find the area of a region bounded by curves, it is essential to first understand the equations of the curves involved. In this case, the curves are given by x = y(y - 1) and y = x/3. Identifying the points of intersection between these curves will help determine the limits of integration for calculating the area.

Integration

Integration is a fundamental concept in calculus used to calculate the area under curves. Once the region is defined by the intersection points, the area can be found by setting up an integral. Depending on the orientation of the curves, this may involve integrating with respect to x or y, which requires understanding how to express the functions appropriately.

Area Between Curves

The area between two curves can be calculated by integrating the difference of the functions that define the upper and lower bounds of the region. In this case, once the curves are expressed in a suitable form, the area can be computed by integrating the difference between the two functions over the interval defined by their points of intersection.

Find the area of the region described in the following exercises.The region bounded by x=y(y−1) and y=x/3