Answer :
Answer:
79.64
Step-by-step explanation:
Consider the boundary curve map
Since
v = 0 == T(u, 0) = (u^2, 0), the line y = 0
v = u == T(u, u) = (0, 2u^2), the line x = 0
Since
u = 2 == T(2, v) = (4 - v^2, 4v).
Since x = 4 - v^2 and y = 4v,
V=y/4
eliminating t yields x = 4 - (y/4)^2 ==> y^2 = 64 - 16x.
Hence T transforms the triangle to D.
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Since x = u^2 - v^2 and y = 2uv,
∂(x,y)/∂(u,v) =
|2u -2v|
|2v 2u| = 4(u^2 + v^2).
Therefore, change of variables yields
∫∫D √(x^2 + y^2) dx dy
= ∫(u = 0 to 4) ∫(v = 0 to u) √[(u^2 - v^2)^2 + (2uv)^2] * 4(u^2 + v^2) dv du
= ∫(u = 0 to 2) ∫(v = 0 to u) √(u^2 + v^2)^2 * 4(u^2 + v^2) dv du
= ∫(u = 0 to 2) ∫(v = 0 to u) 4(u^2 + v^2)^2 dv du
= ∫(u = 0 to 2) ∫(v = 0 to u) 4(u^4 + 2u^2 v^2 + v^4) dv du
= ∫(u = 0 to 2) 4(u^4 v + 2u^2 v^3/3 + v^5/5) {for v = 0 to u} du
=4((u^5 + 2u^5/3 + u^5/5) - (0))
= ∫(u = 0 to 4) (112/15) u^5) du
= (112/15) u^6/6 {for u = 0 to 2}
= 112 * 2^6/90
= 7166/90 = 358/45 = 79.64
The mathematical theorem which is relevant to this problem is called the Variable Change theorem or formula.
What is the Variable Change Form theorem?
The Variable Change Theorem is a theorem that explains how lengths, areas, volumes, and generalized n-dimensional volumes are skewed by functions that can be differentiated.
The step-by-step solution is given as follows
Information given about the boundary curve map is such that:
V = 0 = T(u, 0) = (U², 0), the line y = 0
V = U == T(u, u) = (0, 2U²), the line x = 0
Recall that:
U = 2 = T(2, v) = (4 - V², 4V).
If x = 4 - V² and y = 4V, V = Y/4
Eliminating t will become:
X = 4 - (y/4)² → y² = 64 - 16x.
It, therefore, derives that T transforms the triangle to D.
Because x = U² - V² and Y = 2UV, we can say
∂(x,y)/∂(u,v) = |2u -2v|
|2v 2u| = 4(U² + V²).
Hence the
∫∫D √(x² + y²) dx dy
= ∫(u = 0 to 4) ∫(v = 0 to u) √[(u² - v²)² + (2uv)²] * 4(u² + v²) dv du
= ∫(u = 0 to 2) ∫(v = 0 to u) √(u² + v²)² * 4(u² + v²) dv du
= ∫(u = 0 to 2) ∫(v = 0 to u) 4(u² + v²)² dv du
= ∫(u = 0 to 2) ∫(v = 0 to u) 4(u⁴ + 2u² v² + v⁴) dv du
= ∫(u = 0 to 2) 4(u⁴ v + 2u⁴ v³/3 + v⁵/5) {for v = 0 to u} du
=4((u⁵ + 2u⁵/3 + u⁵/5) - (0))
= ∫(u = 0 to 4) (112/15) u⁵) du
= (112/15) u⁶/6 {for u = 0 to 2}
= 112 * ((2⁶)/90)
= 112 * 0.71111111111
= 79.6444444443
≈ 79.6
Learn more about Variable Change Theorem at
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