4.2 Definite Integrals using Geometry

Front

What is a definite integral?

Back

A definite integral is a mathematical concept that represents the signed area under a curve defined by a function f(x) between two points a and b on the x-axis.

Front

How do you interpret the area under a curve using definite integrals?

Back

The area under a curve from x = a to x = b can be interpreted as the accumulation of quantities represented by the function f(x) over that interval.

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What is the Fundamental Theorem of Calculus?

Back

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

Front

What does it mean for a function to be continuous on an interval?

Back

A function is continuous on an interval if there are no breaks, jumps, or holes in the graph of the function over that interval.

Front

How can geometric shapes help in calculating definite integrals?

Back

Geometric shapes such as rectangles, triangles, and trapezoids can be used to approximate the area under a curve, which can then be refined using limits to find the exact value of the definite integral.

Front

What is the area of a rectangle?

Back

The area of a rectangle is calculated by multiplying its length by its width (Area = length × width).

Front

What is the area of a triangle?

Back

The area of a triangle is calculated using the formula: Area = 1/2 × base × height.

Front

What is the trapezoidal rule?

Back

The trapezoidal rule is a numerical method for estimating the definite integral of a function by dividing the area under the curve into trapezoids and summing their areas.

Front

How do you find the area between two curves?

Back

To find the area between two curves f(x) and g(x) from x = a to x = b, you calculate the integral of the top curve minus the bottom curve: Area = ∫_a^b (f(x) - g(x)) dx.

Front

What is the significance of the limits of integration in a definite integral?

Back

The limits of integration (a and b) define the interval over which the area under the curve is calculated, indicating the starting and ending points on the x-axis.