To determine whether the function f(x) = cos(√x√2 - x^2) is Riemann integrable on the interval [0, 1], we need to check if it satisfies the necessary conditions for Riemann integrability.

1. Boundedness: We need to check if the function is bounded on the interval [0, 1]. Since the cosine function is bounded between -1 and 1 for any input, we can conclude that f(x) is bounded on [0, 1].

2. Discontinuities: We need to check if the function has any discontinuities on the interval [0, 1]. The function f(x) is a composition of continuous functions, namely the square root function and the cosine function. Both of these functions are continuous on their respective domains. Therefore, f(x) is also continuous on [0, 1] and does not have any discontinuities.

Since f(x) is bounded and continuous on [0, 1], it satisfies the necessary conditions for Riemann integrability. Therefore, we can conclude that the function f(x) = cos(√x√2 - x^2) is Riemann integrable on the interval [0, 1].

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To determine whether the function f(x) = cos(√x√2 - x^2) is Riemann integrable on the interval [0, 1], we need to check if it satisfies the necessary conditions for Riemann integrability.

1. Boundedness: We need to check if the function is bounded on the interval [0, 1]. Since the cosine function is bounded between -1 and 1 for any input, we can conclude that f(x) is bounded on [0, 1].

2. Discontinuities: We need to check if the function has any discontinuities on the interval [0, 1]. The function f(x) is a composition of continuous functions, namely the square root function and the cosine function. Both of these functions are continuous on their respective domains. Therefore, f(x) is also continuous on [0, 1] and does not have any discontinuities.

Since f(x) is bounded and continuous on [0, 1], it satisfies the necessary conditions for Riemann integrability. Therefore, we can conclude that the function f(x) = cos(√x√2 - x^2) is Riemann integrable on the interval [0, 1].

Knowee AI · Accepted Answer

To determine whether the function f(x) = cos(√x√2 - x^2) is Riemann integrable on the interval [0, 1], we need to check if it satisfies the necessary conditions for Riemann integrability.

1. Boundedness: We need to check if the function is bounded on the interval [0, 1]. Since the cosine function is bounded between -1 and 1 for any input, we can conclude that f(x) is bounded on [0, 1].

2. Discontinuities: We need to check if the function has any discontinuities on the interval [0, 1]. The function f(x) is a composition of continuous functions, namely the square root function and the cosine function. Both of these functions are continuous on their respective domains. Therefore, f(x) is also continuous on [0, 1] and does not have any discontinuities.

Since f(x) is bounded and continuous on [0, 1], it satisfies the necessary conditions for Riemann integrability. Therefore, we can conclude that the function f(x) = cos(√x√2 - x^2) is Riemann integrable on the interval [0, 1].

Determine whether the following functions are Riemann integrable on [0, 1] or not.(1) f (x) := cos √x√2 − x2

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