Definite Integrals

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What is a definite integral?

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A definite integral is a mathematical concept that represents the signed area under a curve defined by a function over a specific interval [a, b]. It is denoted as ∫_a^b f(x) dx.

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What is the Midpoint Riemann Sum?

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The Midpoint Riemann Sum is a method for approximating the definite integral of a function by dividing the interval into subintervals, using the midpoint of each subinterval to evaluate the function.

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How do you calculate the area under a curve using the Trapezoidal Rule?

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The Trapezoidal Rule approximates the area under a curve by dividing it into trapezoids, calculating the area of each trapezoid, and summing them up. The formula is: A ≈ (b-a)/n * (f(a) + 2Σf(x_i) + f(b)) where n is the number of subintervals.

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What is the formula for the area between a curve and the x-axis?

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The area between a curve y = f(x) and the x-axis from x=a to x=b is given by the definite integral: A = ∫_a^b f(x) dx, where f(x) is non-negative.

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What does it mean if a definite integral evaluates to zero?

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If a definite integral evaluates to zero, it indicates that the areas above and below the x-axis cancel each other out over the interval [a, b].

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What is the significance of the Fundamental Theorem of Calculus?

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The Fundamental Theorem of Calculus links differentiation and integration, stating that if F is an antiderivative of f on [a, b], then ∫_a^b f(x) dx = F(b) - F(a).

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How do you find the average value of a function over an interval?

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The average value of a function f(x) over the interval [a, b] is given by the formula: Average = (1/(b-a)) * ∫_a^b f(x) dx.

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What is the relationship between the area under a curve and the definite integral?

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The area under a curve from x=a to x=b is represented by the definite integral ∫_a^b f(x) dx, which calculates the net area considering the sign of f(x).

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What is the Trapezoidal Rule formula?

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The Trapezoidal Rule formula for approximating the integral is: A ≈ (b-a)/2n * (f(a) + 2Σf(x_i) + f(b)), where n is the number of subintervals.

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What is the purpose of using subintervals in Riemann sums?

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Subintervals in Riemann sums allow for a more accurate approximation of the area under a curve by evaluating the function at multiple points within the interval.