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Chapter 34: Problem 900

If \(u=\iint e^{4 x} \cdot y^{3} \cdot d y \cdot d x\), find \(u\)

Short Answer

Expert verified
The short version of the answer is: \( u = \frac{y^{4}e^{4x}}{16} + C \)

Step by step solution

01

Integrate with respect to y

We will first integrate the expression \(e^{4x}\cdot y^{3}\, dy\). The result of this integration will be a function of \(x\). \[ \int e^{4x} \cdot y^{3} \, dy = e^{4x}\int y^{3} \, dy\] Applying power rule for integration, we get: \[= e^{4x} \cdot \frac{y^{4}}{4} + C(x)\] Now, we have a function of x, which will be further integrated with respect to \(x\).
02

Integrate with respect to x

Now we will integrate the obtained function with respect to \(x\). \[ \int \left(e^{4x} \cdot \frac{y^{4}}{4} + C(x)\right) \, dx\] As \(C(x)\) is a constant of integration with respect to \(x\), it will be treated as an arbitrary constant. We'll now integrate the expression, treating \(\frac{y^{4}}{4}\) as a constant: \[= \frac{y^{4}}{4} \int e^{4x} \, dx\] Applying the exponential rule for integration, we get: \[= \frac{y^{4}}{4} \cdot \frac{e^{4x}}{4} + C\] Now, we have the final value for u: \[ u = \frac{y^{4}e^{4x}}{16} + C \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions
Exponential functions are a type of mathematical function where a constant base is raised to the power of a variable exponent. In scientific, engineering, and mathematical applications, they appear frequently because they describe exponential growth or decay, such as population growth or radioactive decay. In our problem, we work with the exponential function expressed as \(e^{4x}\). Here, "\(e\)" represents Euler's number, approximately equal to 2.71828, a fundamental constant in mathematics. When integrating exponential functions, especially those involving \(e\), they maintain their form, which means \(e^{u}\) integrated with respect to \(u\) results in \(e^{u}\) plus a constant of integration. This property makes exponential functions straightforward to manage within integrals.
Power Rule for Integration
The power rule for integration is a fundamental tool used to integrate functions of the form \(x^n\). This rule is especially useful when tackling polynomials or terms where a variable is raised to a power. Formally, it states that \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\), except for \(n = -1\).
In the exercise, we used this rule to integrate \(y^3\). Applying the power rule, we add one to the power of \(y\), making it \(y^4\), and then we divide by this new power, 4.
  • This transforms \(\int y^3 \, dy\) into \(\frac{y^4}{4}\).
  • The constant \(C(x)\) represents an integration constant that depends on the previous integration step. This step helps simplify complex expressions into manageable terms.
Integration with Respect to Variables
In calculus, when we perform integration, it's essential to pay close attention to the variable of integration. This determines what parts of the expression change through the integration process and what parts act as constants. In the given problem, we first integrate with respect to \(y\), treating any term that doesn鈥檛 include \(y\) as a constant.
After this first integration, our function becomes dependent only on \(x\). For the second step, we integrate with respect to \(x\), treating any constants previously solved from \(y\)-integration as unchanging.
  • The distinction between '\(dy\)' and '\(dx\)' is crucial for correct computations.
  • After integrating each part, the result \(\frac{y^4e^{4x}}{16} + C\) visibly reflects how different variables come together in integrals.
Understanding this process helps streamline problem-solving and ensures accuracy in results.

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Most popular questions from this chapter

Evaluate the double integral: \(\| \mathrm{e}^{-\\{(\mathrm{x}) 2+(\mathrm{y}) 2\\}} \mathrm{d} \mathrm{A}\), in the first quadrant and bounded by the circle: \(\mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{a}^{2}\) and the coordinate axes.

Find the area in the first quadrant bounded by the ellipse: \(\mathrm{x}^{2}+4 \mathrm{y}^{2}=5\), and the hyperbola: \(\mathrm{xy}=1\)

Determine the center of gravity of the area bounded by \(y^{2}=2 x, x=2\), and \(y=0\)

Find the area of the first-octant portion of the cylinder \(x^{2}+z^{2}=a^{2}\) that is included between the planes \(y=0\) and \(\mathrm{y}=\mathrm{x}\)

Sketch the volume represented by the iterated integral: \(1 \int_{0} \sqrt{\\{1-(y) 2\\}} \int_{0} \quad 4 y \mathrm{dx} \mathrm{dy}\), and compute its volume.
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