Double Integrals and Area Calculation

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Questions and Answers

What is the first step in finding the area bounded by the curves y²=2x and y=x?

  • Evaluating the double integral
  • Sketching the curves
  • Finding the limits of integration
  • Identifying the points of intersection (correct)

In a double integral with respect to x first and then y, the limits for x go from x = f1(y) to x = f2(y).

True (A)

After finding the inner integral for the horizontal strip when integrating from y=0 to y=2, what expression results?

y - y²/2

The limits of integration for y in the vertical strip are from y = ______ to y = ______.

<p>x, √2x</p> Signup and view all the answers

Match the integration limits with the appropriate function:

<p>y = x = Lower limit for y in horizontal strip y = √2x = Upper limit for y in vertical strip x = y = Upper limit for x in horizontal strip x = y²/2 = Lower limit for x in horizontal strip</p> Signup and view all the answers

What is the result of the outer integral in the horizontal strip example when integrating from y=0 to y=2?

<p>2/3 (D)</p> Signup and view all the answers

The area found using double integrals is always positive.

<p>False (B)</p> Signup and view all the answers

What are the points of intersection for the curves y=√x and y=x-2?

<p>(4, 2) and (0, 0)</p> Signup and view all the answers

The first integration step for the double integral I = ∫∫(1) dy dx is ______.

<p>integrating with respect to y</p> Signup and view all the answers

Which equation represents the horizontal strip approach for the area bounded by the curves y=√x and y=x-2?

<p>I = ∫(from x=0 to x=4) ∫(from y=x-2 to y=√x) (1) dy dx (C)</p> Signup and view all the answers

What are the solutions obtained when substituting y=√x into y=x-2 and simplifying the resulting equation?

<p>x=1 or x=4 (C)</p> Signup and view all the answers

The area under the curve from x=0 to x=4 is evaluated using a single integral.

<p>False (B)</p> Signup and view all the answers

What is the value of I when integrating with respect to y after changing the order of integration for the function e^x?

<p>2e - 2</p> Signup and view all the answers

The intersection points found from the equations y=√x and y=x-2 are (1,1) and (4, ____).

<p>2</p> Signup and view all the answers

Match the following parts of the integration process with their corresponding results:

<p>∫ (from x=0 to x=1) ∫ (from y=-√x to y=√x) (1) dy dx = 2 ∫ (from y=-1 to y=2) ∫ (from x=y² to x=y+2) (1) dx dy = 9/2 ∫ (from y=0 to y=2) ∫ (from x=0 to x=y/2) e^x dx dy = 2e - 2 ∫ (from x=1 to x=4) ∫ (from y=x-2 to y=√x) (1) dy dx = 9/2</p> Signup and view all the answers

What is the lower limit for the first section of integration in the vertical strip approach?

<p>y=-√x (B)</p> Signup and view all the answers

Changing the order of integration simplifies the integration process when the original order is complex.

<p>True (A)</p> Signup and view all the answers

The function evaluated for the horizontal strip is I = ∫∫( _____ ) dx dy.

<p>1</p> Signup and view all the answers

What type of curves do the equations y=-√x and y=√x represent?

<p>Parabolas</p> Signup and view all the answers

After solving the first part of the integral in the vertical strip, what is the result?

<p>2 (C)</p> Signup and view all the answers

Flashcards

Double Integral

A double integral represents the volume under the surface of a function f(x, y) over a region in the xy-plane.

Horizontal Strip

A horizontal strip in a double integral represents integrating first with respect to x and then with respect to y. The limits of integration for x are defined by the curves bounding the region, and the limits for y are the overall range of y values.

Vertical Strip

A vertical strip in a double integral represents integrating first with respect to y and then with respect to x. The limits of integration for y are defined by the curves bounding the region, and the limits for x are the overall range of x values.

Finding Points of Intersection

To find the area bounded by two curves, we set their equations equal to each other and solve for the points of intersection. These points define the limits of integration.

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Area using Double Integrals

To find the area bounded by curves, we integrate the function f(x, y) = 1 over the region defined by the curves.

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Choosing the Right Strip

When choosing between horizontal and vertical strips, select the one that simplifies the integration process. Consider the limits of integration and the form of the equations.

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Limits of Integration

The limits of integration in a double integral determine the region over which the function is being integrated. These limits are crucial for accurately calculating the volume or area.

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Example: Area Bounded by y²=2x & y=x

In this example, the area bounded by the curves y² = 2x and y = x is calculated using a double integral. The region is divided into horizontal strips, and the integral is evaluated with respect to x first and then with respect to y.

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Example: Area Bounded by y=√x & y = x - 2

In this example, the area bounded by the curves y = √x and y = x - 2 is calculated using a double integral. The region is divided into vertical strips, and the integral is evaluated with respect to y first and then with respect to x.

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Double Integration

A method of double integration where the area is divided into smaller rectangles and then summed. It involves integrating over both the x and y directions.

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Solve Double Integrals

The process of evaluating a double integral by first integrating over one variable and then the other. This allows for a step-by-step calculation of the integral.

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Changing Order of Integration

Rearranging the order in which you integrate with respect to the variables. This is done by changing the limits of integration for each variable while keeping the same integrand. It can make the integral easier to solve.

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Changing Limits of Integration

The process of changing the order of integration in a double integral. This involves determining the new limits of integration for each variable based on the geometry of the integration region.

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Solving Changed Integral

Evaluating a double integral after changing the order of integration. This involves setting up the new integral with the correct limits and solving the inner and outer integrals.

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Study Notes

Double Integrals

  • Double integrals are used to find areas, volumes, and other quantities over two-dimensional regions.
  • The notation for a double integral is ∬f(x,y) dA, where f(x,y) represents the function, and dA represents the area element.
  • Double integrals can be evaluated using iterated integrals (integrating one variable at a time).
  • The order of integration affects the calculation; some integrals are easier to solve if the order is changed.
  • Using horizontal or vertical strips can simplify the integral limits.

Finding the Area Bounded by Curves

  • Finding the area bounded between two curves or functions involves determining the points of intersection between the two functions.
  • This is necessary to define the limits of integration for the area calculation.
  • The points of intersection guide the limits of x and y for suitable double integrals evaluating area computations.
  • Vertical or horizontal strips can be utilized in the integral setup.

Change of Order of Integration

  • The order of integration can be switched to potentially simplify calculation, depending on the region and function.
  • This is often necessary to find appropriate integration limits.
  • Switching the order changes the iterated integration process, which usually leads to new integration limits.
  • Changing the order of integration is particularly relevant when integrals have complicated integration limits or functions.

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