Definite Integrals

Answer

A definite integral ∫[a to b] f(x)dx represents the area under the curve y = f(x) between x = a and x = b. As limit of sum: ∫[a to b] f(x)dx = lim[n→∞] (1/n)[f(a) + f(a+h) + ... + f(a+(n-1)h)] where …

Answer

If f is continuous in [a,b] and F is an antiderivative of f in [a,b], then: ∫[a to b] f(x)dx = F(b) - F(a) = [F(x)]ᵇₐ Significance: - Connects differentiation and integration - Provides an easy meth…

Answer

Step-by-step solution: Given: a = 1, b = 2, f(x) = x, h = 1/n ∫[1 to 2] x dx = lim[n→∞] (1/n)[(1) + (1 + 1/n) + ... + (1 + (n-1)/n)] = lim[n→∞] (1/n)[n + (1/n)(1 + 2 + ... + (n-1))] = lim[n→∞] (1/…

Answer

This property shows that reversing the limits of integration changes the sign of the integral. Explanation: - When we swap upper and lower limits, the integral becomes negative - This maintains consi…

Answer

Additivity Property: The integral over an interval can be split at any intermediate point. Condition: a < b < c Example: Evaluate ∫[0 to 4] x dx using this property with b = 2: ∫[0 to 4] x dx = ∫[0…

Answer

Proof by substitution: Let x = a - t, then dx = -dt When x = 0, t = a; when x = a, t = 0 ∫[0 to a] f(x)dx = ∫[a to 0] f(a-t)(-dt) = ∫[0 to a] f(a-t)dt = ∫[0 to a] f(a-x)dx Application Example: Evalu…

Answer

For even and odd functions: ∫[-a to a] f(x)dx = { 2∫[0 to a] f(x)dx, if f(x) is even [f(-x) = f(x)] 0, if f(x) is odd [f(-x) = -f(x)] } Example 1 (Even): f(x) = x² ∫[-2 to 2] x² dx = 2∫[0 to 2] …

Answer

Step-by-step solution: Step 1: Choose substitution Let 2x² - 1 = t Step 2: Find dt 4x dx = dt x dx = dt/4 Step 3: Change limits When x = 2: t = 2(4) - 1 = 7 When x = 3: t = 2(9) - 1 = 17 Step 4: T…