The area of D is given by:
The average value of f over D is given by:
![\frac{1}{ \frac{343}{3} } \int\limits^7_0 \int\limits^{x^2}_0 {4x\sin(y)} \, dydx = -\frac{3}{343} \int\limits^7_0 {4x\cos(x^2)} \, dx \\ \\ =-\frac{3}{343} \int\limits^{49}_0 {2\cos(t)} \, dt=-\frac{6}{343} \left[\sin(t)\right]_0^{49} \, dt=-\frac{6}{343}\sin49](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B%20%5Cfrac%7B343%7D%7B3%7D%20%7D%20%20%5Cint%5Climits%5E7_0%20%20%5Cint%5Climits%5E%7Bx%5E2%7D_0%20%7B4x%5Csin%28y%29%7D%20%5C%2C%20dydx%20%20%3D%20-%5Cfrac%7B3%7D%7B343%7D%20%20%5Cint%5Climits%5E7_0%20%7B4x%5Ccos%28x%5E2%29%7D%20%5C%2C%20dx%20%20%5C%5C%20%20%5C%5C%20%3D-%5Cfrac%7B3%7D%7B343%7D%20%5Cint%5Climits%5E%7B49%7D_0%20%7B2%5Ccos%28t%29%7D%20%5C%2C%20dt%3D-%5Cfrac%7B6%7D%7B343%7D%20%5Cleft%5B%5Csin%28t%29%5Cright%5D_0%5E%7B49%7D%20%5C%2C%20dt%3D-%5Cfrac%7B6%7D%7B343%7D%5Csin49)
The product is negative 81 t squared + 16 ⇒ 2nd answer
Step-by-step explanation:
The product of two binomials (ax + b)(cx + d), where a, b, c, and d are constant
- Multiply (ax) by (cx) ⇒ 1st × 1st
- Multiply (ax) by (d) and (b) by (cx) ⇒ ext-reams and nears
- Add the two products ⇒ like terms
- Multiply (b) by (d) ⇒ 2nd × 2nd
Let us find the product of (9 t - 4) and (-9 t - 4)
Multiply the 1st two terms
∵ (9 t)(-9 t) = -81 t²
Multiply the ext-reams
∵ (9 t)(-4) = -36 t
Multiply the nears
∵ (-4)(-9 t) = 36 t
Add the like terms
∵ -36 t + 36 t = 0
Multiply the 2nd two terms
∵ (-4)(-4) = 16
Write the answer
∴ (9 t - 4)(-9 t - 4) = -81 t² + 0 + 16
∴ (9 t - 4)(-9 t - 4) = -81 t² + 16
The product is -81 t² + 16
Learn more:
You can learn more about the product of algebraic expressions in brainly.com/question/1617787
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Answer:
Step-by-step explanation:
Its given:
Answer:
The proof contains a simple direct proof, wrapped inside the unnecessary logical packaging of a proof by contradiction framework.
Step-by-step explanation:
The proof is rigourous and well written, so we discard the second answer.
This is not a fake proof by contradiction: it does not have any logical fallacies (circular arguments) or additional assumptions, like, for example, the "proof" of "All the horses are the same color". It is factually correct, but it can be rewritten as a direct proof.
A meaningful proof by contradiction depends strongly on the assumption that the statement to prove is false. In this argument, we only this assumption once, thus it is innecessary. Other proofs by contradiction, like the proof of "The square root of 2 is irrational" or Euclid's proof of the infinitude of primes, develop a longer argument based on the new assumption, but this proof doesn't.
To rewrite this without the superfluous framework, erase the parts "Suppose that the statement is false" and "The fact that the statement is true contradicts the assumption that the statement is false. Thus, the assumption that the statement was false must have been false. Thus, the statement is true."
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