Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region. x+y^2=42, x+y=0
Why are you switching names ????
Follow the method I showed you in the other two questions of this type.
Did you look at the solutions?
Let me know how far you get.
Yes I figured it out thanks
To sketch the region enclosed by the given curves, we first need to understand the equations of the curves. The two curves given are:
1) x + y^2 = 42
2) x + y = 0
Let's analyze each equation separately:
1) x + y^2 = 42:
This equation represents a parabola that opens to the right. We can rewrite it as y = √(42 - x) or y = -√(42 - x) to make it easier to visualize.
If we plot this equation, we will see a parabola that intersects the x-axis at x = 42 and opens toward the right. The upper half of the curve represents the positive values of y, while the lower half represents the negative values of y.
2) x + y = 0:
This equation represents a straight line with a negative slope. If we rearrange it to y = -x, we can observe that it passes through the origin (0, 0) and has a slope of -1.
If we graph this line, we will see a line that intersects the y-axis at y = 0 and the x-axis at x = 0. It has a negative slope (-1) and passes through the origin.
To find the region enclosed by these curves, we need to determine the points of intersection between the parabola and the line.
Setting the two equations equal to each other, we have:
√(42 - x) = -x
To solve for x, we square both sides of the equation:
42 - x = x^2
Rearranging this quadratic equation, we get:
x^2 + x - 42 = 0
Factoring this quadratic equation, we have:
(x + 7)(x - 6) = 0
Thus, we have two solutions for x:
x = -7 or x = 6
Substituting these values back into the equation x + y = 0 (the line), we can find the corresponding y-values:
For x = -7: -7 + y = 0
y = 7
For x = 6: 6 + y = 0
y = -6
Therefore, the points of intersection between the parabola and line are (-7, 7) and (6, -6).
To determine the integral limits, we need to observe the region enclosed by the curves. Since the line (x + y = 0) is below the parabola (x + y^2 = 42), we can integrate with respect to y.
Hence, the integral limits will be y = -6 to y = 7.
Now, to find the area of the region, we need to evaluate the integral ∫(x_high - x_low) y dy.
Our integral becomes:
∫ [√(42 - x) - (-√(42 - x))] dy
Simplifying this, we get:
∫ [2√(42 - x)] dy
Now, we can integrate with respect to y:
∫ [2√(42 - x)] dy = 2∫√(42 - x) dy
Evaluating this integral with the limits of y = -6 to y = 7, we have:
2∫(√(42 - x)) dy from -6 to 7
Finally, to find the area, we need to evaluate this definite integral.