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Area between curves: $f(y)=y$ and $g(y)=y^2$ ?

Find intersection points: $y = y^2$ => $y = 0, 1$ . 2. Set up integral: $\int_{0}^{1} (y - y^2) , dy$ . 3. Evaluate: $[frac{1}{2}y^2 - frac{1}{3}y^3]_0^1 = frac{1}{6}$ .

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Area between curves: $f(y)=y$ and $g(y)=y^2$ ?

Find intersection points: $y = y^2$ => $y = 0, 1$ . 2. Set up integral: $\int_{0}^{1} (y - y^2) , dy$ . 3. Evaluate: $[frac{1}{2}y^2 - frac{1}{3}y^3]_0^1 = frac{1}{6}$ .

Find area between $x=2y-y^3$ and $x=-y$ .

Graph functions. 2. Find intersection points. 3. Set up integral: $\int_{0}^{1.73} (2y - y^3 - (-y)) , dy$ . 4. Evaluate using calculator: Area ≈ 2.25.

Formula for area between curves (y)?

$A = \int_{c}^{d} |f(y) - g(y)| , dy$

How to find intersection points?

Set the two functions equal to each other and solve for $y$ .

Define definite integral.

The definite integral represents the area between a curve and the x-axis over a specified interval.

What is integration with respect to y?

Integration with respect to y involves finding the area between curves by summing horizontal slices along the y-axis.

Define horizontal slices in area calculation.

Horizontal slices are rectangles used to approximate the area between curves when integrating with respect to y.

What are intersection points?

Points where two or more curves meet; used to determine the limits of integration.

All Flashcards

Area between curves: $f(y)=y$ and $g(y)=y^2$ ?

Find intersection points: $y = y^2$ => $y = 0, 1$ . 2. Set up integral: $\int_{0}^{1} (y - y^2) , dy$ . 3. Evaluate: $[frac{1}{2}y^2 - frac{1}{3}y^3]_0^1 = frac{1}{6}$ .

Find area between $x=2y-y^3$ and $x=-y$ .

Graph functions. 2. Find intersection points. 3. Set up integral: $\int_{0}^{1.73} (2y - y^3 - (-y)) , dy$ . 4. Evaluate using calculator: Area ≈ 2.25.

Formula for area between curves (y)?

$A = \int_{c}^{d} |f(y) - g(y)| , dy$

How to find intersection points?

Set the two functions equal to each other and solve for $y$ .

Define definite integral.

The definite integral represents the area between a curve and the x-axis over a specified interval.

What is integration with respect to y?

Integration with respect to y involves finding the area between curves by summing horizontal slices along the y-axis.

Define horizontal slices in area calculation.

Horizontal slices are rectangles used to approximate the area between curves when integrating with respect to y.

What are intersection points?

Points where two or more curves meet; used to determine the limits of integration.