Definite integral of shifted function | Summary and Q&A

TL;DR

Shifting the bounds of a definite integral by adding or subtracting a constant does not change its value.

Key Insights

• 🪜 Shifting a function horizontally by adding or subtracting a constant is equivalent to shifting the bounds of a definite integral by the same constant.
• 🛟 Shifting the function or bounds of a definite integral preserves the area under the curve.
• 😥 Shifting the function does not alter the integral value, but it changes the points at which the function has specific values.
• ❓ Recognizing the ability to shift the bounds or function of a definite integral can simplify problem-solving and provide insights into complex mathematical situations.

Transcript

• Let's say that we know that this area under the curve, y is equal to f of x, let me label it, y is equal to f of x. So under this curve, above the x axis, between a and b, which we denote is the definite integral from a to b, of f of x, d of x. Let's say we know what this is. Let's say it's equal to, let's say this area is equal to five. So given... Read More

Questions & Answers

Q: What happens when we shift the function f(x) by adding a constant, c?

Shifting the function f(x) to the right or left by adding or subtracting a constant, c, moves the entire graph horizontally without changing its shape. The value of the function at x=c is the same as the value at x=0 in the original function.

Q: How does shifting the bounds of a definite integral impact its value?

Shifting the bounds of a definite integral by the same constant, c, does not change its value. The resulting area under the shifted curve is equal to the original integral value.

Q: Can shifting the function and bounds of a definite integral be useful in solving complex problems?

Yes, identifying the ability to shift the function and bounds of a definite integral can be valuable in solving certain math problems. It allows for simplification and understanding of the problem at hand.

Q: Are there any limitations or special cases where shifting the integral bounds may not hold?

Shifting the integral bounds by a constant applies in most cases. However, if the function has points of discontinuity or other special situations, the shifting property may not hold, and further analysis may be needed.

Summary & Key Takeaways

• The definite integral from a to b of a function, denoted as the area under the curve, is equal to a specific value (e.g., 5).

• Shifting the function f(x) by adding or subtracting a constant, c, shifts the graph to the right or left.

• Shifting the bounds of the definite integral by the same constant, c, results in the same area under the shifted curve, which is equal to the original integral value.