Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate the iterated integrals in Problems \(1-14 .\) $$ \int_{0}^{1} \int_{0}^{3 x} x^{2} d y d x $$

Step-by-Step Solution

Answer

The evaluated iterated integral is \( \frac{3}{4} \).

Step 1: Understand the Iterated Integral

The given iterated integral is \( \int_{0}^{1} \int_{0}^{3x} x^2 \, dy \, dx \). This represents the volume under the surface \( x^2 \) over the region bounded by \( 0 \leq x \leq 1 \) and \( 0 \leq y \leq 3x \). We will integrate with respect to \( y \) first and then \( x \).

Step 2: Integrate with respect to y

The integral with respect to \( y \) is \( \int_{0}^{3x} x^2 \, dy \). Since \( x^2 \) is treated as a constant with respect to \( y \), the integration is straightforward: \[ \int_{0}^{3x} x^2 \, dy = x^2 \cdot y \Big|_0^{3x} = x^2 (3x) - x^2 (0) = 3x^3. \]

Step 3: Integrate with respect to x

Now, we integrate the result from Step 2 with respect to \( x \): \( \int_{0}^{1} 3x^3 \, dx \). This can be calculated as: \[ \int_0^1 3x^3 \, dx = 3 \int_0^1 x^3 \, dx = 3 \left( \frac{x^4}{4} \right) \Big|_0^1 = 3 \cdot \frac{1^4}{4} - 3 \cdot \frac{0^4}{4} = \frac{3}{4}. \]

Step 4: Conclude the Calculation

The evaluation of the iterated integral yields a final result. The volume under the curve \( x^2 \) over the specified region is \( \frac{3}{4} \).

Key Concepts

Multiple Integration

Multiple integration is a powerful tool in calculus that allows you to compute volumes, areas, and other quantities by integrating a function over a region. When you see an iterated integral like \( \int_{0}^{1} \int_{0}^{3x} x^2 \, dy \, dx \), you are essentially performing a double integration. This means that you integrate a function multiple times, each with respect to a different variable.

In the exercise above, we are first integrating with respect to \( y \) while treating \( x \) as a constant, and then integrating with respect to \( x \). This order of integration can sometimes be rearranged if the limits allow, but you'll need to be cautious about the geometry of the region of integration.

• **Inner Integral**: Integrate with respect to \( y \), treating \( x \) as constant. This simplification makes it easy to handle, just like we simplify expressions in algebra. Here, since \( x^2 \) is constant with respect to \( y \), the integration with respect to \( y \) becomes straightforward.
• **Outer Integral**: After integrating with respect to \( y \), we move to the outer integral which is with respect to \( x \). This may often involve more complex functions, polynomial integration, or even trigonometric functions, demanding a good grasp of integral rules.

Volume Under a Curve

One of the primary applications of iterated integrals in calculus is to find the volume under a curve over a specific region. The integral \( \int_{0}^{1} \int_{0}^{3x} x^2 \, dy \, dx \) represents the volume under the surface described by \( x^2 \) over the set area defined by the limits of \( x \) and \( y \).

The boundaries given in the integral limits effectively describe the physical space over which the volume exists. It's kind of like slicing a loaf of bread where each slice is the volume contributed by a tiny part of the domain.

• **Region of Integration**: The limits \( 0 \leq x \leq 1 \) and \( 0 \leq y \leq 3x \) define a triangle in the xy-plane. This means the volume is calculated below the curve \( x^2 \) restricted by the triangled boundaries.
• **Geometric Interpretation**: The result of this iterated integral \( \frac{3}{4} \) gives a concise numerical description of the three-dimensional space occupied under that section of the curve. It's like describing a quantifiable portion of the space under terrain with hills and valleys.

Calculus Problem Solving

Solving problems in calculus, particularly with multiple integrals, requires systematic steps and logical thinking. It's a process from analysis to synthesis and conclusion.

Understanding iterated integrals involves examining each part of the integral equation and performing calculations in a step-by-step manner:

• **Analyze the Integral**: Examine the function to be integrated \( (x^2) \) and its limits. Understand what each part of the integral represents in terms of area or volume.
• **Perform the Computations**: Carry out integrations carefully. Start with the innermost integral to simplify and make calculations manageable. Use fundamental integration rules and techniques you’ve learned.
• **Synthesize the Results**: After each integration, consolidate your results. This may involve plugging values back into the equation or summarizing findings in terms of volume or area.
• **Validate and Conclude**: Double-check calculations by confirming that the integrated bounds and resulting computations make logical sense. In this exercise, arriving at \( \frac{3}{4} \) illustrates a deeper understanding of the physical volume represented.

Approaching calculus problems with strategic methods not only simplifies the task but also deepens your comprehension of how mathematical principles and applications intertwine.