Answer: ∫c ((3y + 7e^(√x) dx + (8x + 5 cos (y²)) dy) = ∫∫ₐ 5dA = 5/3
Step-by-step explanation:
Given that;
∫c ((3y + 7e^(√x) dx + (8x + 5 cos (y²)) dy)
Green's Theorem is given as;
∫c (P(x,y)dx + Q(x,y)dy) = ∫∫ₐ { (-β/βy) P(x,y) + (β/βy) Q(x,y) } dA
Now our P(x,y) = 3y + 7e^(√x) and our Q(x,y) = 8x + 5 cos (y²)
Since we know this, therefore; we substitute
∫c ((3y + 7e^(√x) dx + (8x + 5 cos (y²)) dy) = ∫∫ₐ { (-β/βy) (3y + 7e^(√x)) + (β/βy) (8x + 5 cos (y²)) } dA
∫c ((3y + 7e^(√x) dx + (8x + 5 cos (y²)) dy) = ∫∫ₐ ( 8-3) dA
∫c ((3y + 7e^(√x) dx + (8x + 5 cos (y²)) dy) = ∫∫ₐ 5dA
from the question, our region is defined by a lower bound: y = x² and an upper bound of y = √x
going from x = 0 to x = 1
Now calculating ∫∫ₐ 5dA by means of the description of the region, we say;
∫∫ₐ 5dA = 5¹∫₀ ₓ²∫^(√x) dydx
∫∫ₐ 5dA = 5¹∫₀ (y)∧(y-√x) ∨(y-x²) dx
∫∫ₐ 5dA = 5¹∫₀ (√x-x²) dx
∫∫ₐ 5dA = 5 [ ((x^(3/2))/(3/2)) - x³/3]¹₀ NOW since ∫[f(x)]ⁿ dx = ([f(x)]ⁿ⁺¹ / n+1) + C
then
∫∫ₐ 5dA = 5 [ ((1^(3/2))/(3/2)) - 1³ / 3) - ((0^(3/2))/(3/2)) - 0³ / 3) ]
∫∫ₐ 5dA = 5 [ ((1^(3/2))/(3/2)) - 1³ / 3)
∫∫ₐ 5dA = 5/3
Therefore ∫c ((3y + 7e^(√x) dx + (8x + 5 cos (y²)) dy) = ∫∫ₐ 5dA = 5/3
In this exercise we have to use green's theorem to calculate the values of the curve through the integers, so we will find that:
[tex]\int\limits \int\limits_a{5} \, dA = 5/3[/tex]
First, the integral given in this exercise corresponds to:
[tex]\int\limits_C {((3y+7e{\sqrt{x}} dx) + (8x+5cos(y^2)) } \, dy[/tex]
Green's Theorem is given as;
[tex]\int\limits_C {(P(x,y)dx + Q(x,y)dy)} \, = \int\limits \int\limits_a { (-\beta/ \beta_y) P(x,y) + (\beta/ \beta_y) Q(x,y) } dA \,[/tex]
Now our:
[tex]P(x,y) = 3y + 7e^{(\sqrt{x} )} \\ Q(x,y) = 8x + 5 cos (y^2)[/tex]
Since we know this, therefore; we substitute:
[tex]\int\limits_C {((3y + 7e^{(\sqrt{x} )} dx + (8x + 5 cos (y^2)) dy)} = \int\limits \int\limits_a {(-\beta / \beta_y) (3y + 7e^{(\sqrt{x} )} + (\beta / \beta_y) (8x + 5 cos (y^2))} \, dA\\\int\limits_C {((3y + 7e^{(\sqrt{x} )}dx + (8x + 5 cos (y^2)) dy} = \int\limits \int\limits_a {( 8-3) dA}\\\int\limits_C {((3y + 7e^{(\sqrt{x} )}dx + (8x + 5 cos (y^2)) dy} = \int\limits \int\limits_a {5 dA}\\[/tex]
From the question, our region is defined by:
Now calculating by means of the description of the region, we say;
[tex]\int\limits \int\limits_a {5} \, dA = 5 \int\limits^1_0 \int\limits^{\sqrt{x} }_{x^2} {x^2} \, dydx\\\int\limits \int\limits_a {5} \, dA = 5 \int\limits^1_0 {y^{\sqrt{x} } \, dx \\\int\limits \int\limits_a {5} \, dA = 5 \int\limits^1_0 {{\sqrt{x} -x^2} \, dx\\\int\limits \int\limits_a {5} \, dA = 5 [ ((x^{3/2})/(3/2)) - x^3/3][/tex]
Knowing the formula:
[tex]\int\limits {[f(x)]^n} \, dx = ([f(x)]^{n+1} / n+1) + C[/tex]
Applying the values in the presented formula we find that:
[tex]\int\limits \int\limits_a{5} \, dA = 5/3[/tex]
See more about Green's Theorem at brainly.com/question/15062695
