Consider the area between the graphs x+4y=14 and x+7=y^2. This area can be
computed in two different ways using integrals.
First of all it can be computed as a sum of two integrals
They ask to use two integrals so i put f(x) from -7 to 2 which is correct
but for g(x) i put 2 to 14 for some reason 14 is wrong. I also put
f(x)=sqrt(x+7) and g(x)= (14-x)/4 and both are wrong wrong. I got
everything else correct except for these and I don't what I did wrong.
first, graph the two equations (in one cartesian plane)
then get the points of intersection:
x+7=y^2 *this is the second equation
x=y^2-7
substitute this to the first:
y^2-7+4y=14
y^2+4y-21=0
(y+7)(y-3)=0
y=-7 and y=3
substitute these back to obtain corresponding values of x:
*if y=-7,
x=(-7)^2-7=42
*if y=3,
x=(3)^2-7=2
therefore, points of int are (42,-7) and (2,3)
looking at the graph, i suggest you do vertical strips (that is, dx),, divide the whole area into region 1 (left side) and region 2 (right side),, after you do this, get the boundaries.
Region 1:
for x: the boundaries are the graph of the parabola (that is x=y^2-7) and the x-coord of the first point of int (x=2)
for y: since parabola is symmetric with respect to x-axis, y is from -3 to 3.
the area of region 1 is:
integral[from -3 to 3](integral[from y^2-7 to 2] dx)dy)
*note: this is double integral since A=dxdy
Region 2:
for region 2, i suggest you do horizontal strips (that is, dy) ,,then get the boundaries:
for x: from x-coord of first point of int (x=2) to x-ccord of 2nd point of int (x=42)
for y: from the parabola (y=sqrt(x+7)) to the line (y=(14-x)/4)
the area of region 2 is:
integral[from 2 to 42](integral[from sqrt(x+7) to (14-x)/4] dy)dx)
i'll leave the integration calculation to you,, add the areas and you'll finally get the whole area.
so there,, sorry for long explanation,,
i hope i was able to help,, =)
To find the area between the graphs x+4y=14 and x+7=y^2, we can use two different approaches, each involving a separate integral.
First, let's set up the problem using the "vertical slices" approach:
1. Solve both equations for y to express them as functions of x:
x + 4y = 14 => y = (14 - x) / 4
x + 7 = y^2 => y = √(x + 7)
2. Determine the interval of integration by finding the x-values at which the two curves intersect:
Solve (14 - x) / 4 = √(x + 7) to find the x-values of intersection points.
(14 - x) / 4 = √(x + 7)
Square both sides: (14 - x)^2 / 16 = x + 7
Simplify: (196 - 28x + x^2) / 16 - x - 7 = 0
Rearrange: x^2 - 45x + 105 = 0
Use the quadratic formula to solve for x, and you'll find two roots: x = 2 and x = 11.
Therefore, our interval of integration is from x = 2 to x = 11.
3. Set up the integral for the area between the curves:
The integral will be the sum of the areas of the vertical slices between the curves.
Let's denote the upper curve as g(x) and the lower curve as f(x).
integral [from 2 to 11] of (g(x) - f(x)) dx
4. Determine the correct expressions for g(x) and f(x):
From step 1, we already found:
g(x) = (14 - x) / 4
f(x) = √(x + 7)
Therefore, the integral should be:
integral [from 2 to 11] of [(14 - x) / 4 - √(x + 7)] dx
Now, the second approach is using the "horizontal slices" method. Let's go through it:
1. Solve both equations for x to express them as functions of y:
x + 4y = 14 => x = 14 - 4y
x + 7 = y^2 => x = y^2 - 7
2. Determine the interval of integration by finding the y-values at which the two curves intersect:
Solve 14 - 4y = y^2 - 7 to find the y-values of intersection points.
Rearrange y^2 + 4y - 21 = 0
Factorize: (y + 7)(y - 3) = 0
The two roots are y = -7 and y = 3.
Therefore, our interval of integration is from y = -7 to y = 3.
3. Set up the integral for the area between the curves:
The integral will be the sum of the areas of the horizontal slices between the curves.
Let's denote the right curve as g(y) and the left curve as f(y).
integral [from -7 to 3] of (g(y) - f(y)) dy
4. Determine the correct expressions for g(y) and f(y):
From step 1, we already found:
g(y) = y^2 - 7
f(y) = (14 - 4y) / 4 = (7 - 2y) / 2
Therefore, the integral should be:
integral [from -7 to 3] of [(y^2 - 7) - ((7 - 2y) / 2)] dy
Make sure to evaluate both integrals separately and sum the results to find the total area between the curves.
To compute the area between the graphs x+4y=14 and x+7=y^2 using two integrals, let's break it into two separate regions.
First, let's find the x-values where the two curves intersect.
Setting x+4y=14 equal to x+7-y^2, we get:
x + 4y = x + 7 - y^2
4y + y^2 = 7
y^2 + 4y - 7 = 0
Factoring or using the quadratic formula, we find the solutions: y = (-4 ± √40)/2 = -2 ± √10.
Therefore, the two curves intersect at y = -2 - √10 and y = -2 + √10.
Now, let's calculate the area of the first region:
1. Set up the integral for the first region:
∫[a, b] f(x) dx, where f(x) is the upper curve and [a, b] is the x-interval where the two curves intersect.
For the upper curve, we have x+7 = y^2.
Solving for x, we get: x = y^2 - 7.
So, the integral for the first region is:
∫[-2 - √10, -2 + √10] (y^2 - 7) dy.
2. Calculate the integral:
∫[-2 - √10, -2 + √10] (y^2 - 7) dy
= ∫[-2 - √10, -2 + √10] (y^2 dy - 7 dy).
Integrating y^2 with respect to y, we get y^3/3. Integrating 7 with respect to y, we get 7y.
So, the integral becomes:
[ y^3/3 - 7y ] from -2 - √10 to -2 + √10.
Evaluating the integral at the upper and lower limits, we get:
( ((-2 + √10)^3)/3 - 7(-2 + √10) ) - ( ((-2 - √10)^3)/3 - 7(-2 - √10) ).
This gives you the value of the first integral, which represents the area of the first region between the curves.
Now, let's move to the second region:
3. Set up the integral for the second region:
∫[c, d] g(x) dx, where g(x) is the lower curve and [c, d] is the x-interval where the two curves intersect.
For the lower curve, we have x + 4y = 14.
Solving for y, we get: y = (14 - x)/4.
So, the integral for the second region is:
∫[c, d] (14 - x)/4 dx.
4. Calculate the integral:
∫[c, d] (14 - x)/4 dx
= (1/4) ∫[c, d] (14 - x) dx.
Integrating (14 - x) with respect to x, we get 14x - (x^2/2).
So, the integral becomes:
(1/4) [ 14x - (x^2/2) ] from c to d.
Evaluating the integral at the upper and lower limits, we get:
(1/4) [ 14d - (d^2/2) - (14c - (c^2/2)) ].
This gives you the value of the second integral, representing the area of the second region between the curves.
Finally, to compute the total area between the two curves, add the values of the two integrals:
Total area = first integral + second integral.
Make sure to substitute the correct values of a, b, c, and d in your calculations.
If you're still having trouble, please provide the specific values you used for a, b, c, and d, and I can further assist you in identifying the error.