Problem 36

Question

The area lying in the first quadrant inside the circle \(x^{2}+y^{2}=12\) and bounded by the parabolas \(y^{2}=4 x\), \(x^{2}=4 y\) is (A) \(2\left(\frac{\sqrt{2}}{3}+\frac{3}{2} \sin ^{-1} \frac{1}{3}\right)\) (B) \(4\left(\frac{\sqrt{2}}{3}+\frac{3}{2} \sin ^{-1} \frac{1}{3}\right)\) (C) \(\left(\frac{\sqrt{2}}{3}+\frac{3}{2} \sin ^{-1} \frac{1}{3}\right)\) (D) None of these

Step-by-Step Solution

Verified

Answer

The answer is (C): \(\frac{\sqrt{2}}{3} + \frac{3}{2} \sin^{-1}\frac{1}{3}\).

1Step 1: Identify the region of interest

We need to find the area in the first quadrant inside the circle \(x^2 + y^2 = 12\), and bounded by the parabolas \(y^2 = 4x\) and \(x^2 = 4y\). This means solving for the area between these curves in the first quadrant.

2Step 2: Solve for intersection points

The circle intersects with the parabola \(y^2 = 4x\) where \(x^2 + y^2 = 12\) and \(y^2 = 4x\). Substitute \(y^2 = 4x\) into the circle's equation: \(x^2 + 4x = 12\). For the parabola \(x = \frac{y^2}{4}\), substitute into the circle to also find intersections.

3Step 3: Area calculation

Calculate the area bounded by the curve using integration. Substitute \(y = \sqrt{4x}\) into circle \(x^2 + y^2 = 12\) and integrate accordingly in the desired quadrant to calculate the definite integral.

4Step 4: Check and simplify results

After integration, simplify the result to compare with the given options. Ensure all trigonometric and algebraic conversions provide simplified expressions, and the numeric calculations are exact.

Key Concepts

CircleParabolaArea Calculation

Circle

A circle is a perfectly symmetrical geometric figure, where all points are an equal distance from the center point. This distance is known as the radius. The equation for a circle centered at the origin is given by \(x^2 + y^2 = r^2\), where \(r\) is the radius. In our problem, the circle \(x^2 + y^2 = 12\) has a radius of \(\sqrt{12}\) or approximately 3.46.
Key aspects of a circle include:

Symmetry: Circles are symmetric with respect to their center.
Radius: The distance from the center to any point on the circle.
Circumference: The total distance around the circle, calculated as \(2\pi r\).
Area: The space contained within the circle, which is \(\pi r^2\).

In analytical geometry, circles often form part of the boundaries within which areas or intersections need to be calculated. Here, the circle forms a boundary that restricts the region of interest within the first quadrant when intersected by the parabolas provided in the exercise.

Parabola

Parabolas are symmetrical open curves with a unique U-shape, defined as the set of all points equidistant from a fixed point known as a focus and a line called the directrix. In the exercise, we deal with two parabolas: \(y^2 = 4x\) and \(x^2 = 4y\).
For the parabola \(y^2 = 4x\):

It opens to the right, based on the squared variable being \(y^2\).
The vertex is at the origin \((0,0)\).
The focus is point \((1,0)\) as calculated from the standard form \(y^2 = 4ax\), where \(a = 1\).

For the parabola \(x^2 = 4y\):

It opens upward since \(x^2\) is on the left-hand side.
The vertex is also at the origin \((0,0)\).
The focus is point \((0,1)\), calculated from \(x^2 = 4ay\) with \(a = 1\).

These parabolas intersect in the first quadrant, creating a bounded region from which the desired area is to be calculated. Understanding the orientation and geometry of these parabolas is crucial for solving the task at hand.

Area Calculation

The problem requires calculating the area in the first quadrant that is inside the circle and bounded by the given parabolas. This process involves finding the area of complex regions using calculus techniques such as integration.
Here is how you approach the calculation:

**Identify Boundaries:** Understand which curves form the boundaries of the area of interest. In this case, the circle and the two parabolas provide the necessary boundaries.
**Intersection Points:** Calculate where these curves intersect in the first quadrant. Substituting the equations of parabolas into the circle's equation, solve for the intersection points.
**Setup Integrals:** Use integrals to find the areas between intersections. Integration here will involve definite integrals over the specific arms formed between intersections and within the circle's radius.
**Accounting for Symmetry:** As the calculation resides in the first quadrant, it takes advantage of the symmetry of the circle and parabolas around the axes.

The process of integrating and simplifying the result allows comparison against the provided options, ensuring the area computed accurately describes the intersected region. Area calculation like this is essential in analytical geometry as it combines geometrical understanding with algebraic manipulation.

Problem 35

Problem 37

Other exercises in this chapter

Problem 34

Let \(f\) beintegrable over \([0, a]\) for any real \(a\). If we define \(I_{1}=\int_{0}^{\pi / 2} \cos \theta f\left(\sin \theta+\cos ^{2} \theta\right) d \the

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Problem 35

The area bounded by the circle \(x^{2}+y^{2}=8\), the parabola \(x^{2}=2 y\) and the line \(y=x\) in \(y \geq 0\) is (A) \(\frac{2}{3}+2 \pi\) (B) \(\frac{2}{3}

View solution

Problem 37

If \(f(y)=e^{y}, g(y)=y ; y>0\) and \(\phi(t)=\int_{0}^{t} f(t-y) g(y) d y\), then \(\phi(t)=\) (A) \(e^{t}-(1+t)\) (B) \(1-e^{-t}(1+t)\) (C) \(t e^{t}\) (D) No

View solution

Problem 38

The value of \(\int_{-1}^{1} \frac{\sin ^{2} x}{\left[\frac{x}{\sqrt{2}}\right]+\frac{1}{2}} d x\), where \([x]=\) greatest integer less than or equal to \(x\),

View solution