Equal integrals Without evaluating integrals, explain the following equalities. (Hint: Draw pictures.)


b. ∫²₀(25−(x²+1)²) dx = 2∫₁⁵ y√y−1 dy

Verified step by step guidance

1. Step 1: Observe the integral on the left-hand side, ∫₂₀(25−(x²+1)²) dx. This represents the area under the curve defined by the function f(x) = 25−(x²+1)² over the interval [0, 2]. The function is symmetric about the y-axis, and its shape suggests a geometric interpretation.
2. Step 2: Rewrite the integral on the right-hand side, 2∫₁⁵ y√y−1 dy. This represents twice the area under the curve defined by g(y) = y√y−1 over the interval [1, 5]. The factor of 2 indicates symmetry or a doubling of the area.
3. Step 3: Use the hint to draw pictures. For the left-hand side, sketch the curve f(x) = 25−(x²+1)², which is a parabola-like shape inverted due to the negative sign. For the right-hand side, sketch the curve g(y) = y√y−1, which is a function involving a square root and grows as y increases.
4. Step 4: Analyze the symmetry and transformations. The integral on the left-hand side can be interpreted geometrically as the area of a region that corresponds to the integral on the right-hand side after a change of variables. Specifically, the substitution x²+1 ↔ y and adjustments to limits of integration align the two integrals.
5. Step 5: Conclude that the equality holds because the two integrals represent the same geometric area under their respective curves, albeit expressed in different coordinate systems. The factor of 2 in the right-hand side accounts for symmetry or doubling of the area.

Key Concepts

Definite Integrals

Definite integrals represent the signed area under a curve between two points on the x-axis. They are calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. Understanding the properties of definite integrals, such as linearity and symmetry, is crucial for analyzing equalities involving integrals without direct evaluation.

Substitution Method

The substitution method is a technique used in integration to simplify the process by changing variables. This method involves substituting a part of the integrand with a new variable, which can make the integral easier to evaluate. Recognizing when and how to apply substitution is essential for understanding the relationship between different integrals, as seen in the given equality.

Geometric Interpretation of Integrals

Integrals can be interpreted geometrically as areas under curves or between curves. Visualizing the functions involved in the integrals can reveal relationships and symmetries that may not be immediately apparent through algebraic manipulation. Drawing diagrams helps in understanding how the areas represented by the integrals relate to each other, which is key to explaining the given equality.