Question

Use double integrals to compute the area of the following regions. Make a sketch of the region. The region bounded by the parabola \(y=x^{2}\) and the line \(y=4\)

Short answer

Question: Use double integrals to compute the area of the region bounded by the parabola \(y = x^2\) and the line \(y = 4\). Answer: The area of the region is \(\frac{32}{3}\) square units.

Step-by-step solution

Sketch the Region and Find Intersection Points

Start by sketching the parabola \(y = x^2\) and the line \(y = 4\) on the same coordinate plane. Notice that the line \(y = 4\) represents a horizontal line at a height of 4 units above the x-axis, and the parabola is a quadratic curve that opens upward. To find the points of intersection between these two curves, set \(y = x^2\) and \(y = 4\) equal to each other: \(x^2 = 4\) Solve for x: \(x = \pm \sqrt{4}\) \(x = \pm 2\) So, the points of intersection are \((-2, 4)\) and \((2, 4)\). These will be used to set up the limits of integration for the double integral.

Set Up the Double Integral

The area of the region we are trying to find can be represented as a double integral, which measures the volume under the curve \(z = 1\) over the region in the xy-plane. The integral will be in the form: \(\iint_{R}{1 \, dy \, dx}\) where R is the region bounded by the parabola \(y = x^2\) and the line \(y = 4\). For the x-limits of integration, the region extends from \(x = -2\) to \(x = 2\). For the y-limits, we will integrate with respect to the y-axis, starting from the parabola and moving up to the line. So, the y-variable ranges from the parabola (in terms of x): \(y = x^2\) to the line \(y = 4\). Thus, the double integral becomes: \(\int_{-2}^{2}\int_{x^2}^{4}{1 \, dy \, dx}\)

Compute the Double Integral

Evaluate the double integral by solving the inner integral with respect to y first: \(\int_{-2}^{2}{(4 - x^2) \, dx}\) Now, integrate with respect to x: \(\int_{-2}^{2}{4 - x^2 \, dx} = \big[4x - \frac{1}{3}x^3\big]_{-2}^{2}\) Evaluate the expression at the limits of integration: \(= \left[4(2) - \frac{1}{3}(2^3)\right] - \left[4(-2) - \frac{1}{3}(-2^3)\right]\) Simplify: \(= (8 - \frac{8}{3}) - (-8 + \frac{8}{3})\) \(= \frac{16}{3} + \frac{16}{3}\) \(= \frac{32}{3}\) So, the area of the region bounded by the parabola and the line is \(\frac{32}{3}\) square units.

Key concepts

Calculus

Calculus is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. This vast field plays a crucial role in everything from scientific research to economic theories and engineering. It has two primary branches: differential calculus, concerning rates of change and slopes of curves, and integral calculus, which is focused on the accumulation of quantities and the areas under and between curves.

Understanding calculus is fundamental for tackling more complex mathematical problems, especially those involving changes and motion. It's the gateway to not only advance math but also physics, engineering, computer science, statistics, and more.

Integration

Integration, a core concept within calculus, is the process of finding the integral of a function, which represents the area under a curve on a graph. There are two main types of integrals: indefinite and definite. An indefinite integral represents a family of functions and includes a constant of integration. A definite integral, which is used to calculate the exact area under a curve between specified limits, has upper and lower bounds and does not include the constant.

Learning how to integrate functions is essential when attempting to find volumes, areas, and other quantities when the geometry is not straightforward. It's used extensively in multiple disciplines, from environmental science modeling the growth of populations to economics finding consumer and producer surplus.

Multivariable Calculus

Multivariable calculus is an extension of single-variable calculus that includes functions of several variables. It's where the concept of a double integral arises. Double integrals extend the idea of a definite integral to functions of two variables, such as calculating the volume under a surface or the area of a region in the xy-plane.

The subject expands on the principles of calculus to three dimensions and beyond, allowing us to tackle problems involving multiple rates of change and complex systems. A strong grasp of multivariable calculus is necessary for fields ranging from 3D graphics and meteorology to advanced physics and optimization problems in engineering.

Definite Integrals

Definite integrals are integrals with upper and lower limits, which give a specific numerical value. They are essential for computing the accumulation of a function over an interval. For example, this can mean finding the total distance traveled by an object given its velocity over time, or in our problem, the area beneath a curve or between curves.

Definite integrals serve as a precise tool for quantifying enclosed area, and mastering their computation is key for problem-solving in many scientific and engineering scenarios.

Area of a Region

The concept of the area of a region in calculus is pivotal when needing to determine the space inside a curve or between curves. In multivariable calculus, this is commonly calculated with double integrals. Double integrals work by accumulating infinitesimally small rectangles over the entire region. Efficiently setting the proper bounds for these integrals requires understanding the geometry involved, such as curves or surfaces that bound the region.

Being adept at calculating the area of a region is not only useful in pure mathematics but also in applications such as determining land areas for real estate, the cross-sectional area in engineering designs, or the graphic representation of statistical data.